qrencode.lua

The qrcode library is licensed under the 3-clause BSD license (aka "new BSD") To get in contact with the author, mail to gundlach@speedata.de.

Please report bugs on the github project page.

-- Copyright (c) 2012, Patrick Gundlach
-- All rights reserved.
--
-- Redistribution and use in source and binary forms, with or without
-- modification, are permitted provided that the following conditions are met:
--	 * Redistributions of source code must retain the above copyright
--	   notice, this list of conditions and the following disclaimer.
--	 * Redistributions in binary form must reproduce the above copyright
--	   notice, this list of conditions and the following disclaimer in the
--	   documentation and/or other materials provided with the distribution.
--	 * Neither the name of the <organization> nor the
--	   names of its contributors may be used to endorse or promote products
--	   derived from this software without specific prior written permission.
--
-- THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
-- ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
-- WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
-- DISCLAIMED. IN NO EVENT SHALL <COPYRIGHT HOLDER> BE LIABLE FOR ANY
-- DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
-- (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
-- LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
-- ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
-- (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
-- SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Overall workflow

The steps to generate the qrcode, assuming we already have the codeword:

  1. Determine version, ec level and mode (=encoding) for codeword
  2. Encode data
  3. Arrange data and calculate error correction code
  4. Generate 8 matrices with different masks and calculate the penalty
  5. Return qrcode with least penalty

Each step is of course more or less complex and needs further description

Helper functions

We start with some helper functions

-- To calculate xor we need to do that bitwise. This helper table speeds up the num-to-bit
-- part a bit (no pun intended)
local cclxvi = {[0] = {0,0,0,0,0,0,0,0}, {1,0,0,0,0,0,0,0}, {0,1,0,0,0,0,0,0}, {1,1,0,0,0,0,0,0},
{0,0,1,0,0,0,0,0}, {1,0,1,0,0,0,0,0}, {0,1,1,0,0,0,0,0}, {1,1,1,0,0,0,0,0},
{0,0,0,1,0,0,0,0}, {1,0,0,1,0,0,0,0}, {0,1,0,1,0,0,0,0}, {1,1,0,1,0,0,0,0},
{0,0,1,1,0,0,0,0}, {1,0,1,1,0,0,0,0}, {0,1,1,1,0,0,0,0}, {1,1,1,1,0,0,0,0},
{0,0,0,0,1,0,0,0}, {1,0,0,0,1,0,0,0}, {0,1,0,0,1,0,0,0}, {1,1,0,0,1,0,0,0},
{0,0,1,0,1,0,0,0}, {1,0,1,0,1,0,0,0}, {0,1,1,0,1,0,0,0}, {1,1,1,0,1,0,0,0},
{0,0,0,1,1,0,0,0}, {1,0,0,1,1,0,0,0}, {0,1,0,1,1,0,0,0}, {1,1,0,1,1,0,0,0},
{0,0,1,1,1,0,0,0}, {1,0,1,1,1,0,0,0}, {0,1,1,1,1,0,0,0}, {1,1,1,1,1,0,0,0},
{0,0,0,0,0,1,0,0}, {1,0,0,0,0,1,0,0}, {0,1,0,0,0,1,0,0}, {1,1,0,0,0,1,0,0},
{0,0,1,0,0,1,0,0}, {1,0,1,0,0,1,0,0}, {0,1,1,0,0,1,0,0}, {1,1,1,0,0,1,0,0},
{0,0,0,1,0,1,0,0}, {1,0,0,1,0,1,0,0}, {0,1,0,1,0,1,0,0}, {1,1,0,1,0,1,0,0},
{0,0,1,1,0,1,0,0}, {1,0,1,1,0,1,0,0}, {0,1,1,1,0,1,0,0}, {1,1,1,1,0,1,0,0},
{0,0,0,0,1,1,0,0}, {1,0,0,0,1,1,0,0}, {0,1,0,0,1,1,0,0}, {1,1,0,0,1,1,0,0},
{0,0,1,0,1,1,0,0}, {1,0,1,0,1,1,0,0}, {0,1,1,0,1,1,0,0}, {1,1,1,0,1,1,0,0},
{0,0,0,1,1,1,0,0}, {1,0,0,1,1,1,0,0}, {0,1,0,1,1,1,0,0}, {1,1,0,1,1,1,0,0},
{0,0,1,1,1,1,0,0}, {1,0,1,1,1,1,0,0}, {0,1,1,1,1,1,0,0}, {1,1,1,1,1,1,0,0},
{0,0,0,0,0,0,1,0}, {1,0,0,0,0,0,1,0}, {0,1,0,0,0,0,1,0}, {1,1,0,0,0,0,1,0},
{0,0,1,0,0,0,1,0}, {1,0,1,0,0,0,1,0}, {0,1,1,0,0,0,1,0}, {1,1,1,0,0,0,1,0},
{0,0,0,1,0,0,1,0}, {1,0,0,1,0,0,1,0}, {0,1,0,1,0,0,1,0}, {1,1,0,1,0,0,1,0},
{0,0,1,1,0,0,1,0}, {1,0,1,1,0,0,1,0}, {0,1,1,1,0,0,1,0}, {1,1,1,1,0,0,1,0},
{0,0,0,0,1,0,1,0}, {1,0,0,0,1,0,1,0}, {0,1,0,0,1,0,1,0}, {1,1,0,0,1,0,1,0},
{0,0,1,0,1,0,1,0}, {1,0,1,0,1,0,1,0}, {0,1,1,0,1,0,1,0}, {1,1,1,0,1,0,1,0},
{0,0,0,1,1,0,1,0}, {1,0,0,1,1,0,1,0}, {0,1,0,1,1,0,1,0}, {1,1,0,1,1,0,1,0},
{0,0,1,1,1,0,1,0}, {1,0,1,1,1,0,1,0}, {0,1,1,1,1,0,1,0}, {1,1,1,1,1,0,1,0},
{0,0,0,0,0,1,1,0}, {1,0,0,0,0,1,1,0}, {0,1,0,0,0,1,1,0}, {1,1,0,0,0,1,1,0},
{0,0,1,0,0,1,1,0}, {1,0,1,0,0,1,1,0}, {0,1,1,0,0,1,1,0}, {1,1,1,0,0,1,1,0},
{0,0,0,1,0,1,1,0}, {1,0,0,1,0,1,1,0}, {0,1,0,1,0,1,1,0}, {1,1,0,1,0,1,1,0},
{0,0,1,1,0,1,1,0}, {1,0,1,1,0,1,1,0}, {0,1,1,1,0,1,1,0}, {1,1,1,1,0,1,1,0},
{0,0,0,0,1,1,1,0}, {1,0,0,0,1,1,1,0}, {0,1,0,0,1,1,1,0}, {1,1,0,0,1,1,1,0},
{0,0,1,0,1,1,1,0}, {1,0,1,0,1,1,1,0}, {0,1,1,0,1,1,1,0}, {1,1,1,0,1,1,1,0},
{0,0,0,1,1,1,1,0}, {1,0,0,1,1,1,1,0}, {0,1,0,1,1,1,1,0}, {1,1,0,1,1,1,1,0},
{0,0,1,1,1,1,1,0}, {1,0,1,1,1,1,1,0}, {0,1,1,1,1,1,1,0}, {1,1,1,1,1,1,1,0},
{0,0,0,0,0,0,0,1}, {1,0,0,0,0,0,0,1}, {0,1,0,0,0,0,0,1}, {1,1,0,0,0,0,0,1},
{0,0,1,0,0,0,0,1}, {1,0,1,0,0,0,0,1}, {0,1,1,0,0,0,0,1}, {1,1,1,0,0,0,0,1},
{0,0,0,1,0,0,0,1}, {1,0,0,1,0,0,0,1}, {0,1,0,1,0,0,0,1}, {1,1,0,1,0,0,0,1},
{0,0,1,1,0,0,0,1}, {1,0,1,1,0,0,0,1}, {0,1,1,1,0,0,0,1}, {1,1,1,1,0,0,0,1},
{0,0,0,0,1,0,0,1}, {1,0,0,0,1,0,0,1}, {0,1,0,0,1,0,0,1}, {1,1,0,0,1,0,0,1},
{0,0,1,0,1,0,0,1}, {1,0,1,0,1,0,0,1}, {0,1,1,0,1,0,0,1}, {1,1,1,0,1,0,0,1},
{0,0,0,1,1,0,0,1}, {1,0,0,1,1,0,0,1}, {0,1,0,1,1,0,0,1}, {1,1,0,1,1,0,0,1},
{0,0,1,1,1,0,0,1}, {1,0,1,1,1,0,0,1}, {0,1,1,1,1,0,0,1}, {1,1,1,1,1,0,0,1},
{0,0,0,0,0,1,0,1}, {1,0,0,0,0,1,0,1}, {0,1,0,0,0,1,0,1}, {1,1,0,0,0,1,0,1},
{0,0,1,0,0,1,0,1}, {1,0,1,0,0,1,0,1}, {0,1,1,0,0,1,0,1}, {1,1,1,0,0,1,0,1},
{0,0,0,1,0,1,0,1}, {1,0,0,1,0,1,0,1}, {0,1,0,1,0,1,0,1}, {1,1,0,1,0,1,0,1},
{0,0,1,1,0,1,0,1}, {1,0,1,1,0,1,0,1}, {0,1,1,1,0,1,0,1}, {1,1,1,1,0,1,0,1},
{0,0,0,0,1,1,0,1}, {1,0,0,0,1,1,0,1}, {0,1,0,0,1,1,0,1}, {1,1,0,0,1,1,0,1},
{0,0,1,0,1,1,0,1}, {1,0,1,0,1,1,0,1}, {0,1,1,0,1,1,0,1}, {1,1,1,0,1,1,0,1},
{0,0,0,1,1,1,0,1}, {1,0,0,1,1,1,0,1}, {0,1,0,1,1,1,0,1}, {1,1,0,1,1,1,0,1},
{0,0,1,1,1,1,0,1}, {1,0,1,1,1,1,0,1}, {0,1,1,1,1,1,0,1}, {1,1,1,1,1,1,0,1},
{0,0,0,0,0,0,1,1}, {1,0,0,0,0,0,1,1}, {0,1,0,0,0,0,1,1}, {1,1,0,0,0,0,1,1},
{0,0,1,0,0,0,1,1}, {1,0,1,0,0,0,1,1}, {0,1,1,0,0,0,1,1}, {1,1,1,0,0,0,1,1},
{0,0,0,1,0,0,1,1}, {1,0,0,1,0,0,1,1}, {0,1,0,1,0,0,1,1}, {1,1,0,1,0,0,1,1},
{0,0,1,1,0,0,1,1}, {1,0,1,1,0,0,1,1}, {0,1,1,1,0,0,1,1}, {1,1,1,1,0,0,1,1},
{0,0,0,0,1,0,1,1}, {1,0,0,0,1,0,1,1}, {0,1,0,0,1,0,1,1}, {1,1,0,0,1,0,1,1},
{0,0,1,0,1,0,1,1}, {1,0,1,0,1,0,1,1}, {0,1,1,0,1,0,1,1}, {1,1,1,0,1,0,1,1},
{0,0,0,1,1,0,1,1}, {1,0,0,1,1,0,1,1}, {0,1,0,1,1,0,1,1}, {1,1,0,1,1,0,1,1},
{0,0,1,1,1,0,1,1}, {1,0,1,1,1,0,1,1}, {0,1,1,1,1,0,1,1}, {1,1,1,1,1,0,1,1},
{0,0,0,0,0,1,1,1}, {1,0,0,0,0,1,1,1}, {0,1,0,0,0,1,1,1}, {1,1,0,0,0,1,1,1},
{0,0,1,0,0,1,1,1}, {1,0,1,0,0,1,1,1}, {0,1,1,0,0,1,1,1}, {1,1,1,0,0,1,1,1},
{0,0,0,1,0,1,1,1}, {1,0,0,1,0,1,1,1}, {0,1,0,1,0,1,1,1}, {1,1,0,1,0,1,1,1},
{0,0,1,1,0,1,1,1}, {1,0,1,1,0,1,1,1}, {0,1,1,1,0,1,1,1}, {1,1,1,1,0,1,1,1},
{0,0,0,0,1,1,1,1}, {1,0,0,0,1,1,1,1}, {0,1,0,0,1,1,1,1}, {1,1,0,0,1,1,1,1},
{0,0,1,0,1,1,1,1}, {1,0,1,0,1,1,1,1}, {0,1,1,0,1,1,1,1}, {1,1,1,0,1,1,1,1},
{0,0,0,1,1,1,1,1}, {1,0,0,1,1,1,1,1}, {0,1,0,1,1,1,1,1}, {1,1,0,1,1,1,1,1},
{0,0,1,1,1,1,1,1}, {1,0,1,1,1,1,1,1}, {0,1,1,1,1,1,1,1}, {1,1,1,1,1,1,1,1}}

-- Return a number that is the result of interpreting the table tbl (msb first)
local function tbl_to_number(tbl)
	local n = #tbl
	local rslt = 0
	local power = 1
	for i = 1, n do
		rslt = rslt + tbl[i]*power
		power = power*2
	end
	return rslt
end

-- Calculate bitwise xor of bytes m and n. 0 <= m,n <= 256.
local function bit_xor(m, n)
	local tbl_m = cclxvi[m]
	local tbl_n = cclxvi[n]
	local tbl = {}
	for i = 1, 8 do
		if(tbl_m[i] ~= tbl_n[i]) then
			tbl[i] = 1
		else
			tbl[i] = 0
		end
	end
	return tbl_to_number(tbl)
end

-- Return the binary representation of the number x with the width of `digits`.
local function binary(x,digits)
  local s=string.format("%o",x)
  local a={["0"]="000",["1"]="001", ["2"]="010",["3"]="011",
		   ["4"]="100",["5"]="101", ["6"]="110",["7"]="111"}
  s=string.gsub(s,"(.)",function (d) return a[d] end)
  -- remove leading 0s
  s = string.gsub(s,"^0*(.*)$","%1")
  local fmtstring = string.format("%%%ds",digits)
  local ret = string.format(fmtstring,s)
  return string.gsub(ret," ","0")
end

-- A small helper function for add_typeinfo_to_matrix() and add_version_information()
-- Add a 2 (black by default) / -2 (blank by default) to the matrix at position x,y
-- depending on the bitstring (size 1!) where "0"=blank and "1"=black.
local function fill_matrix_position(matrix,bitstring,x,y)
	if bitstring == "1" then
		matrix[x][y] = 2
	else
		matrix[x][y] = -2
	end
end

Step 1: Determine version, ec level and mode for codeword

First we need to find out the version (= size) of the QR code. This depends on the input data (the mode to be used), the requested error correction level (normally we use the maximum level that fits into the minimal size).

-- Return the mode for the given string `str`.
-- See table 2 of the spec. We only support mode 1, 2 and 4.
-- That is: numeric, alaphnumeric and binary.
local function get_mode( str )
	local mode
	if string.match(str,"^[0-9]+$") then
		return 1
	elseif string.match(str,"^[0-9A-Z $%%*./:+-]+$") then
		return 2
	else
		return 4
	end
	assert(false,"never reached")
	return nil
end


Capacity of QR codes

The capacity is calculated as follow: \(\text{Number of data bits} = \text{number of codewords} * 8\). The number of data bits is now reduced by 4 (the mode indicator) and the length string, that varies between 8 and 16, depending on the version and the mode (see method get_length()). The remaining capacity is multiplied by the amount of data per bit string (numeric: 3, alphanumeric: 2, other: 1) and divided by the length of the bit string (numeric: 10, alphanumeric: 11, binary: 8, kanji: 13). Then the floor function is applied to the result: $$\Big\lfloor \frac{( \text{#data bits} - 4 - \text{length string}) * \text{data per bit string}}{\text{length of the bit string}} \Big\rfloor$$

There is one problem remaining. The length string depends on the version, and the version depends on the length string. But we take this into account when calculating the the capacity, so this is not really a problem here.

-- The capacity (number of codewords) of each version (1-40) for error correction levels 1-4 (LMQH).
-- The higher the ec level, the lower the capacity of the version. Taken from spec, tables 7-11.
local capacity = {
  {  19,   16,   13,	9},{  34,   28,   22,   16},{  55,   44,   34,   26},{  80,   64,   48,   36},
  { 108,   86,   62,   46},{ 136,  108,   76,   60},{ 156,  124,   88,   66},{ 194,  154,  110,   86},
  { 232,  182,  132,  100},{ 274,  216,  154,  122},{ 324,  254,  180,  140},{ 370,  290,  206,  158},
  { 428,  334,  244,  180},{ 461,  365,  261,  197},{ 523,  415,  295,  223},{ 589,  453,  325,  253},
  { 647,  507,  367,  283},{ 721,  563,  397,  313},{ 795,  627,  445,  341},{ 861,  669,  485,  385},
  { 932,  714,  512,  406},{1006,  782,  568,  442},{1094,  860,  614,  464},{1174,  914,  664,  514},
  {1276, 1000,  718,  538},{1370, 1062,  754,  596},{1468, 1128,  808,  628},{1531, 1193,  871,  661},
  {1631, 1267,  911,  701},{1735, 1373,  985,  745},{1843, 1455, 1033,  793},{1955, 1541, 1115,  845},
  {2071, 1631, 1171,  901},{2191, 1725, 1231,  961},{2306, 1812, 1286,  986},{2434, 1914, 1354, 1054},
  {2566, 1992, 1426, 1096},{2702, 2102, 1502, 1142},{2812, 2216, 1582, 1222},{2956, 2334, 1666, 1276}}

Return the smallest version for this codeword. If requested_ec_level is supplied, then the ec level (LMQH - 1,2,3,4) must be at least the requested level.

-- mode = 1,2,4,8
local function get_version_eclevel(len,mode,requested_ec_level)
	local local_mode = mode
	if mode == 4 then
		local_mode = 3
	elseif mode == 8 then
		local_mode = 4
	end
	assert( local_mode <= 4 )

	local bytes, bits, digits, modebits, c
	local tab = { {10,9,8,8},{12,11,16,10},{14,13,16,12} }
	local minversion = 40
	local maxec_level = requested_ec_level or 1
	local min,max = 1, 4
	if requested_ec_level and requested_ec_level >= 1 and requested_ec_level <= 4 then
		min = requested_ec_level
		max = requested_ec_level
	end
	for ec_level=min,max do
		for version=1,#capacity do
			bits = capacity[version][ec_level] * 8
			bits = bits - 4 -- the mode indicator
			if version < 10 then
				digits = tab[1][local_mode]
			elseif version < 27 then
				digits = tab[2][local_mode]
			elseif version <= 40 then
				digits = tab[3][local_mode]
			end
			modebits = bits - digits
			if local_mode == 1 then -- numeric
				c = math.floor(modebits * 3 / 10)
			elseif local_mode == 2 then -- alphanumeric
				c = math.floor(modebits * 2 / 11)
			elseif local_mode == 3 then -- binary
				c = math.floor(modebits * 1 / 8)
			else
				c = math.floor(modebits * 1 / 13)
			end
			if c >= len then
				if version <= minversion then
					minversion = version
					maxec_level = ec_level
				end
				break
			end
		end
	end
	return minversion, maxec_level
end

-- Return a bit string of 0s and 1s that includes the length of the code string.
-- The modes are numeric = 1, alphanumeric = 2, binary = 4, and japanese = 8
local function get_length(str,version,mode)
	local i = mode
	if mode == 4 then
		i = 3
	elseif mode == 8 then
		i = 4
	end
	assert( i <= 4 )
	local tab = { {10,9,8,8},{12,11,16,10},{14,13,16,12} }
	local digits
	if version < 10 then
		digits = tab[1][i]
	elseif version < 27 then
		digits = tab[2][i]
	elseif version <= 40 then
		digits = tab[3][i]
	else
		assert(false, "get_length, version > 40 not supported")
	end
	local len = binary(#str,digits)
	return len
end

If the requested_ec_level or the mode are provided, this will be used if possible. The mode depends on the characters used in the string str. It seems to be possible to split the QR code to handle multiple modes, but we don't do that.

local function get_version_eclevel_mode_bistringlength(str,requested_ec_level,mode)
	local local_mode
	if mode then
		assert(false,"not implemented")
		-- check if the mode is OK for the string
		local_mode = mode
	else
		local_mode = get_mode(str)
	end
	local version, ec_level
	version, ec_level = get_version_eclevel(#str,local_mode,requested_ec_level)
	local length_string = get_length(str,version,local_mode)
	return version,ec_level,binary(local_mode,4),local_mode,length_string
end

Step 2: Encode data

There are several ways to encode the data. We currently support only numeric, alphanumeric and binary. We already chose the encoding (a.k.a. mode) in the first step, so we need to apply the mode to the codeword.

Numeric: take three digits and encode them in 10 bits Alphanumeric: take two characters and encode them in 11 bits Binary: take one octet and encode it in 8 bits

local asciitbl = {
	    -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  -- 0x01-0x0f
	-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  -- 0x10-0x1f
	36, -1, -1, -1, 37, 38, -1, -1, -1, -1, 39, 40, -1, 41, 42, 43,  -- 0x20-0x2f
	 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 44, -1, -1, -1, -1, -1,  -- 0x30-0x3f
	-1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,  -- 0x40-0x4f
	25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, -1, -1, -1, -1, -1,  -- 0x50-0x5f
  }

-- Return a binary representation of the numeric string `str`. This must contain only digits 0-9.
local function encode_string_numeric(str)
	local bitstring = ""
	local int
	string.gsub(str,"..?.?",function(a)
		int = tonumber(a)
		if #a == 3 then
			bitstring = bitstring .. binary(int,10)
		elseif #a == 2 then
			bitstring = bitstring .. binary(int,7)
		else
			bitstring = bitstring .. binary(int,4)
		end
	end)
	return bitstring
end

-- Return a binary representation of the alphanumeric string `str`. This must contain only
-- digits 0-9, uppercase letters A-Z, space and the following chars: $%*./:+-.
local function encode_string_ascii(str)
	local bitstring = ""
	local int
	local b1, b2
	string.gsub(str,"..?",function(a)
		if #a == 2 then
			b1 = asciitbl[string.byte(string.sub(a,1,1))]
			b2 = asciitbl[string.byte(string.sub(a,2,2))]
			int = b1 * 45 + b2
			bitstring = bitstring .. binary(int,11)
		else
			int = asciitbl[string.byte(a)]
			bitstring = bitstring .. binary(int,6)
		end
	  end)
	return bitstring
end

-- Return a bitstring representing string str in binary mode.
-- We don't handle UTF-8 in any special way because we assume the
-- scanner recognizes UTF-8 and displays it correctly.
local function encode_string_binary(str)
	local ret = {}
	string.gsub(str,".",function(x)
		ret[#ret + 1] = binary(string.byte(x),8)
	end)
	return table.concat(ret)
end

-- Return a bitstring representing string str in the given mode.
local function encode_data(str,mode)
	if mode == 1 then
		return encode_string_numeric(str)
	elseif mode == 2 then
		return encode_string_ascii(str)
	elseif mode == 4 then
		return encode_string_binary(str)
	else
		assert(false,"not implemented yet")
	end
end

-- Encoding the codeword is not enough. We need to make sure that
-- the length of the binary string is equal to the number of codewords of the version.
local function add_pad_data(version,ec_level,data)
	local count_to_pad, missing_digits
	local cpty = capacity[version][ec_level] * 8
	count_to_pad = math.min(4,cpty - #data)
	if count_to_pad > 0 then
		data = data .. string.rep("0",count_to_pad)
	end
	if math.fmod(#data,8) ~= 0 then
		missing_digits = 8 - math.fmod(#data,8)
		data = data .. string.rep("0",missing_digits)
	end
	assert(math.fmod(#data,8) == 0)
	-- add "11101100" and "00010001" until enough data
	while #data < cpty do
		data = data .. "11101100"
		if #data < cpty then
			data = data .. "00010001"
		end
	end
	return data
end


Step 3: Organize data and calculate error correction code

The data in the qrcode is not encoded linearly. For example code 5-H has four blocks, the first two blocks contain 11 codewords and 22 error correction codes each, the second block contain 12 codewords and 22 ec codes each. We just take the table from the spec and don't calculate the blocks ourself. The table ecblocks contains this info.

During the phase of splitting the data into codewords, we do the calculation for error correction codes. This step involves polynomial division. Find a math book from school and follow the code here :)

Reed Solomon error correction

Now this is the slightly ugly part of the error correction. We start with log/antilog tables

local alpha_int = {
	[0] = 0,
	  2,   4,   8,  16,  32,  64, 128,  29,  58, 116, 232, 205, 135,  19,  38,  76,
	152,  45,  90, 180, 117, 234, 201, 143,   3,   6,  12,  24,  48,  96, 192, 157,
	 39,  78, 156,  37,  74, 148,  53, 106, 212, 181, 119, 238, 193, 159,  35,  70,
	140,   5,  10,  20,  40,  80, 160,  93, 186, 105, 210, 185, 111, 222, 161,  95,
	190,  97, 194, 153,  47,  94, 188, 101, 202, 137,  15,  30,  60, 120, 240, 253,
	231, 211, 187, 107, 214, 177, 127, 254, 225, 223, 163,  91, 182, 113, 226, 217,
	175,  67, 134,  17,  34,  68, 136,  13,  26,  52, 104, 208, 189, 103, 206, 129,
	 31,  62, 124, 248, 237, 199, 147,  59, 118, 236, 197, 151,  51, 102, 204, 133,
	 23,  46,  92, 184, 109, 218, 169,  79, 158,  33,  66, 132,  21,  42,  84, 168,
	 77, 154,  41,  82, 164,  85, 170,  73, 146,  57, 114, 228, 213, 183, 115, 230,
	209, 191,  99, 198, 145,  63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227,
	219, 171,  75, 150,  49,  98, 196, 149,  55, 110, 220, 165,  87, 174,  65, 130,
	 25,  50, 100, 200, 141,   7,  14,  28,  56, 112, 224, 221, 167,  83, 166,  81,
	162,  89, 178, 121, 242, 249, 239, 195, 155,  43,  86, 172,  69, 138,   9,  18,
	 36,  72, 144,  61, 122, 244, 245, 247, 243, 251, 235, 203, 139,  11,  22,  44,
	 88, 176, 125, 250, 233, 207, 131,  27,  54, 108, 216, 173,  71, 142,   1
}

local int_alpha = {
	[0] = 0,
	255,   1,  25,   2,  50,  26, 198,   3, 223,  51, 238,  27, 104, 199,  75,   4,
	100, 224,  14,  52, 141, 239, 129,  28, 193, 105, 248, 200,   8,  76, 113,   5,
	138, 101,  47, 225,  36,  15,  33,  53, 147, 142, 218, 240,  18, 130,  69,  29,
	181, 194, 125, 106,  39, 249, 185, 201, 154,   9, 120,  77, 228, 114, 166,   6,
	191, 139,  98, 102, 221,  48, 253, 226, 152,  37, 179,  16, 145,  34, 136,  54,
	208, 148, 206, 143, 150, 219, 189, 241, 210,  19,  92, 131,  56,  70,  64,  30,
	 66, 182, 163, 195,  72, 126, 110, 107,  58,  40,  84, 250, 133, 186,  61, 202,
	 94, 155, 159,  10,  21, 121,  43,  78, 212, 229, 172, 115, 243, 167,  87,   7,
	112, 192, 247, 140, 128,  99,  13, 103,  74, 222, 237,  49, 197, 254,  24, 227,
	165, 153, 119,  38, 184, 180, 124,  17,  68, 146, 217,  35,  32, 137,  46,  55,
	 63, 209,  91, 149, 188, 207, 205, 144, 135, 151, 178, 220, 252, 190,  97, 242,
	 86, 211, 171,  20,  42,  93, 158, 132,  60,  57,  83,  71, 109,  65, 162,  31,
	 45,  67, 216, 183, 123, 164, 118, 196,  23,  73, 236, 127,  12, 111, 246, 108,
	161,  59,  82,  41, 157,  85, 170, 251,  96, 134, 177, 187, 204,  62,  90, 203,
	 89,  95, 176, 156, 169, 160,  81,  11, 245,  22, 235, 122, 117,  44, 215,  79,
	174, 213, 233, 230, 231, 173, 232, 116, 214, 244, 234, 168,  80,  88, 175
}

-- We only need the polynomial generators for block sizes 7, 10, 13, 15, 16, 17, 18, 20, 22, 24, 26, 28, and 30. Version
-- 2 of the qr codes don't need larger ones (as opposed to version 1). The table has the format x^1*ɑ^21 + x^2*a^102 ...
local generator_polynomial = {
	 [7] = { 21, 102, 238, 149, 146, 229,  87,   0},
	[10] = { 45,  32,  94,  64,  70, 118,  61,  46,  67, 251,   0 },
	[13] = { 78, 140, 206, 218, 130, 104, 106, 100,  86, 100, 176, 152,  74,   0 },
	[15] = {105,  99,   5, 124, 140, 237,  58,  58,  51,  37, 202,  91,  61, 183,   8,   0},
	[16] = {120, 225, 194, 182, 169, 147, 191,  91,   3,  76, 161, 102, 109, 107, 104, 120,   0},
	[17] = {136, 163, 243,  39, 150,  99,  24, 147, 214, 206, 123, 239,  43,  78, 206, 139,  43,   0},
	[18] = {153,  96,  98,   5, 179, 252, 148, 152, 187,  79, 170, 118,  97, 184,  94, 158, 234, 215,   0},
	[20] = {190, 188, 212, 212, 164, 156, 239,  83, 225, 221, 180, 202, 187,  26, 163,  61,  50,  79,  60,  17,   0},
	[22] = {231, 165, 105, 160, 134, 219,  80,  98, 172,   8,  74, 200,  53, 221, 109,  14, 230,  93, 242, 247, 171, 210,   0},
	[24] = { 21, 227,  96,  87, 232, 117,   0, 111, 218, 228, 226, 192, 152, 169, 180, 159, 126, 251, 117, 211,  48, 135, 121, 229,   0},
	[26] = { 70, 218, 145, 153, 227,  48, 102,  13, 142, 245,  21, 161,  53, 165,  28, 111, 201, 145,  17, 118, 182, 103,   2, 158, 125, 173,   0},
	[28] = {123,   9,  37, 242, 119, 212, 195,  42,  87, 245,  43,  21, 201, 232,  27, 205, 147, 195, 190, 110, 180, 108, 234, 224, 104, 200, 223, 168,   0},
	[30] = {180, 192,  40, 238, 216, 251,  37, 156, 130, 224, 193, 226, 173,  42, 125, 222,  96, 239,  86, 110,  48,  50, 182, 179,  31, 216, 152, 145, 173, 41, 0}}


-- Turn a binary string of length 8*x into a table size x of numbers.
local function convert_bitstring_to_bytes(data)
	local msg = {}
	local tab = string.gsub(data,"(........)",function(x)
		msg[#msg+1] = tonumber(x,2)
		end)
	return msg
end

-- Return a table that has 0's in the first entries and then the alpha
-- representation of the generator polynominal
local function get_generator_polynominal_adjusted(num_ec_codewords,highest_exponent)
	local gp_alpha = {[0]=0}
	for i=0,highest_exponent - num_ec_codewords - 1 do
		gp_alpha[i] = 0
	end
	local gp = generator_polynomial[num_ec_codewords]
	for i=1,num_ec_codewords + 1 do
		gp_alpha[highest_exponent - num_ec_codewords + i - 1] = gp[i]
	end
	return gp_alpha
end

These converter functions use the log/antilog table above. We could have created the table programatically, but I like fixed tables.

-- Convert polynominal in int notation to alpha notation.
local function convert_to_alpha( tab )
	local new_tab = {}
	for i=0,#tab do
		new_tab[i] = int_alpha[tab[i]]
	end
	return new_tab
end

-- Convert polynominal in alpha notation to int notation.
local function convert_to_int(tab,len_message)
	local new_tab = {}
	for i=0,#tab do
		new_tab[i] = alpha_int[tab[i]]
	end
	return new_tab
end

-- That's the heart of the error correction calculation.
local function calculate_error_correction(data,num_ec_codewords)
	local mp
	if type(data)=="string" then
		mp = convert_bitstring_to_bytes(data)
	elseif type(data)=="table" then
		mp = data
	else
		assert(false,"Unknown type for data: %s",type(data))
	end
	local len_message = #mp

	local highest_exponent = len_message + num_ec_codewords - 1
	local gp_alpha,tmp
	local he
	local gp_int = {}
	local mp_int,mp_alpha = {},{}
	-- create message shifted to left (highest exponent)
	for i=1,len_message do
		mp_int[highest_exponent - i + 1] = mp[i]
	end
	for i=1,highest_exponent - len_message do
		mp_int[i] = 0
	end
	mp_int[0] = 0

	mp_alpha = convert_to_alpha(mp_int)

	while highest_exponent >= num_ec_codewords do
		gp_alpha = get_generator_polynominal_adjusted(num_ec_codewords,highest_exponent)

		-- Multiply generator polynomial by first coefficient of the above polynomial

		-- take the highest exponent from the message polynom (alpha) and add
		-- it to the generator polynom
		local exp = mp_alpha[highest_exponent]
		for i=highest_exponent,highest_exponent - num_ec_codewords,-1 do
			if gp_alpha[i] + exp > 255 then
				gp_alpha[i] = math.fmod(gp_alpha[i] + exp,255)
			else
				gp_alpha[i] = gp_alpha[i] + exp
			end
		end
		for i=highest_exponent - num_ec_codewords - 1,0,-1 do
			gp_alpha[i] = 0
		end

		gp_int = convert_to_int(gp_alpha)
		mp_int = convert_to_int(mp_alpha)


		tmp = {}
		for i=highest_exponent,0,-1 do
			tmp[i] = bit_xor(gp_int[i],mp_int[i])
		end
		-- remove leading 0's
		he = highest_exponent
		for i=he,0,-1 do
			-- We need to stop if the length of the codeword is matched
			if i < num_ec_codewords then break end
			if tmp[i] == 0 then
				tmp[i] = nil
				highest_exponent = highest_exponent - 1
			else
				break
			end
		end
		mp_int = tmp
		mp_alpha = convert_to_alpha(mp_int)
	end
	local ret = {}

	-- reverse data
	for i=#mp_int,0,-1 do
		ret[#ret + 1] = mp_int[i]
	end
	return ret
end

Arranging the data

Now we arrange the data into smaller chunks. This table is taken from the spec.

-- ecblocks has 40 entries, one for each version. Each version entry has 4 entries, for each LMQH
-- ec level. Each entry has two or four fields, the odd files are the number of repetitions for the
-- folowing block info. The first entry of the block is the total number of codewords in the block,
-- the second entry is the number of data codewords. The third is not important.
local ecblocks = {
  {{  1,{ 26, 19, 2}                 },   {  1,{26,16, 4}},                  {  1,{26,13, 6}},                  {  1, {26, 9, 8}               }},
  {{  1,{ 44, 34, 4}                 },   {  1,{44,28, 8}},                  {  1,{44,22,11}},                  {  1, {44,16,14}               }},
  {{  1,{ 70, 55, 7}                 },   {  1,{70,44,13}},                  {  2,{35,17, 9}},                  {  2, {35,13,11}               }},
  {{  1,{100, 80,10}                 },   {  2,{50,32, 9}},                  {  2,{50,24,13}},                  {  4, {25, 9, 8}               }},
  {{  1,{134,108,13}                 },   {  2,{67,43,12}},                  {  2,{33,15, 9},  2,{34,16, 9}},   {  2, {33,11,11},  2,{34,12,11}}},
  {{  2,{ 86, 68, 9}                 },   {  4,{43,27, 8}},                  {  4,{43,19,12}},                  {  4, {43,15,14}               }},
  {{  2,{ 98, 78,10}                 },   {  4,{49,31, 9}},                  {  2,{32,14, 9},  4,{33,15, 9}},   {  4, {39,13,13},  1,{40,14,13}}},
  {{  2,{121, 97,12}                 },   {  2,{60,38,11},  2,{61,39,11}},   {  4,{40,18,11},  2,{41,19,11}},   {  4, {40,14,13},  2,{41,15,13}}},
  {{  2,{146,116,15}                 },   {  3,{58,36,11},  2,{59,37,11}},   {  4,{36,16,10},  4,{37,17,10}},   {  4, {36,12,12},  4,{37,13,12}}},
  {{  2,{ 86, 68, 9},  2,{ 87, 69, 9}},   {  4,{69,43,13},  1,{70,44,13}},   {  6,{43,19,12},  2,{44,20,12}},   {  6, {43,15,14},  2,{44,16,14}}},
  {{  4,{101, 81,10}                 },   {  1,{80,50,15},  4,{81,51,15}},   {  4,{50,22,14},  4,{51,23,14}},   {  3, {36,12,12},  8,{37,13,12}}},
  {{  2,{116, 92,12},  2,{117, 93,12}},   {  6,{58,36,11},  2,{59,37,11}},   {  4,{46,20,13},  6,{47,21,13}},   {  7, {42,14,14},  4,{43,15,14}}},
  {{  4,{133,107,13}                 },   {  8,{59,37,11},  1,{60,38,11}},   {  8,{44,20,12},  4,{45,21,12}},   { 12, {33,11,11},  4,{34,12,11}}},
  {{  3,{145,115,15},  1,{146,116,15}},   {  4,{64,40,12},  5,{65,41,12}},   { 11,{36,16,10},  5,{37,17,10}},   { 11, {36,12,12},  5,{37,13,12}}},
  {{  5,{109, 87,11},  1,{110, 88,11}},   {  5,{65,41,12},  5,{66,42,12}},   {  5,{54,24,15},  7,{55,25,15}},   { 11, {36,12,12},  7,{37,13,12}}},
  {{  5,{122, 98,12},  1,{123, 99,12}},   {  7,{73,45,14},  3,{74,46,14}},   { 15,{43,19,12},  2,{44,20,12}},   {  3, {45,15,15}, 13,{46,16,15}}},
  {{  1,{135,107,14},  5,{136,108,14}},   { 10,{74,46,14},  1,{75,47,14}},   {  1,{50,22,14}, 15,{51,23,14}},   {  2, {42,14,14}, 17,{43,15,14}}},
  {{  5,{150,120,15},  1,{151,121,15}},   {  9,{69,43,13},  4,{70,44,13}},   { 17,{50,22,14},  1,{51,23,14}},   {  2, {42,14,14}, 19,{43,15,14}}},
  {{  3,{141,113,14},  4,{142,114,14}},   {  3,{70,44,13}, 11,{71,45,13}},   { 17,{47,21,13},  4,{48,22,13}},   {  9, {39,13,13}, 16,{40,14,13}}},
  {{  3,{135,107,14},  5,{136,108,14}},   {  3,{67,41,13}, 13,{68,42,13}},   { 15,{54,24,15},  5,{55,25,15}},   { 15, {43,15,14}, 10,{44,16,14}}},
  {{  4,{144,116,14},  4,{145,117,14}},   { 17,{68,42,13}},                  { 17,{50,22,14},  6,{51,23,14}},   { 19, {46,16,15},  6,{47,17,15}}},
  {{  2,{139,111,14},  7,{140,112,14}},   { 17,{74,46,14}},                  {  7,{54,24,15}, 16,{55,25,15}},   { 34, {37,13,12}               }},
  {{  4,{151,121,15},  5,{152,122,15}},   {  4,{75,47,14}, 14,{76,48,14}},   { 11,{54,24,15}, 14,{55,25,15}},   { 16, {45,15,15}, 14,{46,16,15}}},
  {{  6,{147,117,15},  4,{148,118,15}},   {  6,{73,45,14}, 14,{74,46,14}},   { 11,{54,24,15}, 16,{55,25,15}},   { 30, {46,16,15},  2,{47,17,15}}},
  {{  8,{132,106,13},  4,{133,107,13}},   {  8,{75,47,14}, 13,{76,48,14}},   {  7,{54,24,15}, 22,{55,25,15}},   { 22, {45,15,15}, 13,{46,16,15}}},
  {{ 10,{142,114,14},  2,{143,115,14}},   { 19,{74,46,14},  4,{75,47,14}},   { 28,{50,22,14},  6,{51,23,14}},   { 33, {46,16,15},  4,{47,17,15}}},
  {{  8,{152,122,15},  4,{153,123,15}},   { 22,{73,45,14},  3,{74,46,14}},   {  8,{53,23,15}, 26,{54,24,15}},   { 12, {45,15,15}, 28,{46,16,15}}},
  {{  3,{147,117,15}, 10,{148,118,15}},   {  3,{73,45,14}, 23,{74,46,14}},   {  4,{54,24,15}, 31,{55,25,15}},   { 11, {45,15,15}, 31,{46,16,15}}},
  {{  7,{146,116,15},  7,{147,117,15}},   { 21,{73,45,14},  7,{74,46,14}},   {  1,{53,23,15}, 37,{54,24,15}},   { 19, {45,15,15}, 26,{46,16,15}}},
  {{  5,{145,115,15}, 10,{146,116,15}},   { 19,{75,47,14}, 10,{76,48,14}},   { 15,{54,24,15}, 25,{55,25,15}},   { 23, {45,15,15}, 25,{46,16,15}}},
  {{ 13,{145,115,15},  3,{146,116,15}},   {  2,{74,46,14}, 29,{75,47,14}},   { 42,{54,24,15},  1,{55,25,15}},   { 23, {45,15,15}, 28,{46,16,15}}},
  {{ 17,{145,115,15}            	 },   { 10,{74,46,14}, 23,{75,47,14}},   { 10,{54,24,15}, 35,{55,25,15}},   { 19, {45,15,15}, 35,{46,16,15}}},
  {{ 17,{145,115,15},  1,{146,116,15}},   { 14,{74,46,14}, 21,{75,47,14}},   { 29,{54,24,15}, 19,{55,25,15}},   { 11, {45,15,15}, 46,{46,16,15}}},
  {{ 13,{145,115,15},  6,{146,116,15}},   { 14,{74,46,14}, 23,{75,47,14}},   { 44,{54,24,15},  7,{55,25,15}},   { 59, {46,16,15},  1,{47,17,15}}},
  {{ 12,{151,121,15},  7,{152,122,15}},   { 12,{75,47,14}, 26,{76,48,14}},   { 39,{54,24,15}, 14,{55,25,15}},   { 22, {45,15,15}, 41,{46,16,15}}},
  {{  6,{151,121,15}, 14,{152,122,15}},   {  6,{75,47,14}, 34,{76,48,14}},   { 46,{54,24,15}, 10,{55,25,15}},   {  2, {45,15,15}, 64,{46,16,15}}},
  {{ 17,{152,122,15},  4,{153,123,15}},   { 29,{74,46,14}, 14,{75,47,14}},   { 49,{54,24,15}, 10,{55,25,15}},   { 24, {45,15,15}, 46,{46,16,15}}},
  {{  4,{152,122,15}, 18,{153,123,15}},   { 13,{74,46,14}, 32,{75,47,14}},   { 48,{54,24,15}, 14,{55,25,15}},   { 42, {45,15,15}, 32,{46,16,15}}},
  {{ 20,{147,117,15},  4,{148,118,15}},   { 40,{75,47,14},  7,{76,48,14}},   { 43,{54,24,15}, 22,{55,25,15}},   { 10, {45,15,15}, 67,{46,16,15}}},
  {{ 19,{148,118,15},  6,{149,119,15}},   { 18,{75,47,14}, 31,{76,48,14}},   { 34,{54,24,15}, 34,{55,25,15}},   { 20, {45,15,15}, 61,{46,16,15}}}
}

-- The bits that must be 0 if the version does fill the complete matrix.
-- Example: for version 1, no bits need to be added after arranging the data, for version 2 we need to add 7 bits at the end.
local remainder = {0, 7, 7, 7, 7, 7, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0}

-- This is the formula for table 1 in the spec:
-- function get_capacity_remainder( version )
-- 	local len = version * 4 + 17
-- 	local size = len^2
-- 	local function_pattern_modules = 192 + 2 * len - 32 -- Position Adjustment pattern + timing pattern
-- 	local count_alignemnt_pattern = #alignment_pattern[version]
-- 	if count_alignemnt_pattern > 0 then
-- 		-- add 25 for each aligment pattern
-- 		function_pattern_modules = function_pattern_modules + 25 * ( count_alignemnt_pattern^2 - 3 )
-- 		-- but substract the timing pattern occupied by the aligment pattern on the top and left
-- 		function_pattern_modules = function_pattern_modules - ( count_alignemnt_pattern - 2) * 10
-- 	end
-- 	size = size - function_pattern_modules
-- 	if version > 6 then
-- 		size = size - 67
-- 	else
-- 		size = size - 31
-- 	end
-- 	return math.floor(size/8),math.fmod(size,8)
-- end

Example: Version 5-H has four data and four error correction blocks. The table above lists 2, {33,11,11}, 2,{34,12,11} for entry [5][4]. This means we take two blocks with 11 codewords and two blocks with 12 codewords, and two blocks with 33 - 11 = 22 ec codes and another two blocks with 34 - 12 = 22 ec codes.

 Block 1: D1  D2  D3  ... D11
 Block 2: D12 D13 D14 ... D22
 Block 3: D23 D24 D25 ... D33 D34
 Block 4: D35 D36 D37 ... D45 D46

Then we place the data like this in the matrix: D1, D12, D23, D35, D2, D13, D24, D36 ... D45, D34, D46. The same goes with error correction codes.

-- The given data can be a string of 0's and 1' (with #string mod 8 == 0).
-- Alternatively the data can be a table of codewords. The number of codewords
-- must match the capacity of the qr code.
local function arrange_codewords_and_calculate_ec( version,ec_level,data )
	if type(data)=="table" then
		local tmp = ""
		for i=1,#data do
			tmp = tmp .. binary(data[i],8)
		end
		data = tmp
	end
	-- If the size of the data is not enough for the codeword, we add 0's and two special bytes until finished.
	local blocks = ecblocks[version][ec_level]
	local size_datablock_bytes, size_ecblock_bytes
	local datablocks = {}
	local ecblocks = {}
	local count = 1
	local pos = 0
	local cpty_ec_bits = 0
	for i=1,#blocks/2 do
		for j=1,blocks[2*i - 1] do
			size_datablock_bytes = blocks[2*i][2]
			size_ecblock_bytes   = blocks[2*i][1] - blocks[2*i][2]
			cpty_ec_bits = cpty_ec_bits + size_ecblock_bytes * 8
			datablocks[#datablocks + 1] = string.sub(data, pos * 8 + 1,( pos + size_datablock_bytes)*8)
			tmp_tab = calculate_error_correction(datablocks[#datablocks],size_ecblock_bytes)
			tmp_str = ""
			for x=1,#tmp_tab do
				tmp_str = tmp_str .. binary(tmp_tab[x],8)
			end
			ecblocks[#ecblocks + 1] = tmp_str
			pos = pos + size_datablock_bytes
			count = count + 1
		end
	end
	local arranged_data = ""
	pos = 1
	repeat
		for i=1,#datablocks do
			if pos < #datablocks[i] then
				arranged_data = arranged_data .. string.sub(datablocks[i],pos, pos + 7)
			end
		end
		pos = pos + 8
	until #arranged_data == #data
	-- ec
	local arranged_ec = ""
	pos = 1
	repeat
		for i=1,#ecblocks do
			if pos < #ecblocks[i] then
				arranged_ec = arranged_ec .. string.sub(ecblocks[i],pos, pos + 7)
			end
		end
		pos = pos + 8
	until #arranged_ec == cpty_ec_bits
	return arranged_data .. arranged_ec
end

Step 4: Generate 8 matrices with different masks and calculate the penalty

Prepare matrix

The first step is to prepare an empty matrix for a given size/mask. The matrix has a few predefined areas that must be black or blank. We encode the matrix with a two dimensional field where the numbers determine which pixel is blank or not.

The following code is used for our matrix:

 0 = not in use yet,
-2 = blank by mandatory pattern,
 2 = black by mandatory pattern,
-1 = blank by data,
 1 = black by data

To prepare the empty, we add positioning, alingment and timing patters.

Positioning patterns

local function add_position_detection_patterns(tab_x)
	local size = #tab_x
	-- allocate quite zone in the matrix area
	for i=1,8 do
		for j=1,8 do
			tab_x[i][j] = -2
			tab_x[size - 8 + i][j] = -2
			tab_x[i][size - 8 + j] = -2
		end
	end
	-- draw the detection pattern (outer)
	for i=1,7 do
		-- top left
		tab_x[1][i]=2
		tab_x[7][i]=2
		tab_x[i][1]=2
		tab_x[i][7]=2

		-- top right
		tab_x[size][i]=2
		tab_x[size - 6][i]=2
		tab_x[size - i + 1][1]=2
		tab_x[size - i + 1][7]=2

		-- bottom left
		tab_x[1][size - i + 1]=2
		tab_x[7][size - i + 1]=2
		tab_x[i][size - 6]=2
		tab_x[i][size]=2
	end
	-- draw the detection pattern (inner)
	for i=1,3 do
		for j=1,3 do
			-- top left
			tab_x[2+j][i+2]=2
			-- top right
			tab_x[size - j - 1][i+2]=2
			-- bottom left
			tab_x[2 + j][size - i - 1]=2
		end
	end
end

Timing patterns

-- The timing patterns (two) are the dashed lines between two adjacent positioning patterns on row/column 7.
local function add_timing_pattern(tab_x)
	local line,col
	line = 7
	col = 9
	for i=col,#tab_x - 8 do
		if math.fmod(i,2) == 1 then
			tab_x[i][line] = 2
		else
			tab_x[i][line] = -2
		end
	end
	for i=col,#tab_x - 8 do
		if math.fmod(i,2) == 1 then
			tab_x[line][i] = 2
		else
			tab_x[line][i] = -2
		end
	end
end

Alignment patterns

The alignment patterns must be added to the matrix for versions > 1. The amount and positions depend on the versions and are given by the spec. Beware: the patterns must not be placed where we have the positioning patterns (that is: top left, top right and bottom left.)

-- For each version, where should we place the alignment patterns? See table E.1 of the spec
local alignment_pattern = {
  {},{6,18},{6,22},{6,26},{6,30},{6,34}, -- 1-6
  {6,22,38},{6,24,42},{6,26,46},{6,28,50},{6,30,54},{6,32,58},{6,34,62}, -- 7-13
  {6,26,46,66},{6,26,48,70},{6,26,50,74},{6,30,54,78},{6,30,56,82},{6,30,58,86},{6,34,62,90}, -- 14-20
  {6,28,50,72,94},{6,26,50,74,98},{6,30,54,78,102},{6,28,54,80,106},{6,32,58,84,110},{6,30,58,86,114},{6,34,62,90,118}, -- 21-27
  {6,26,50,74,98 ,122},{6,30,54,78,102,126},{6,26,52,78,104,130},{6,30,56,82,108,134},{6,34,60,86,112,138},{6,30,58,86,114,142},{6,34,62,90,118,146}, -- 28-34
  {6,30,54,78,102,126,150}, {6,24,50,76,102,128,154},{6,28,54,80,106,132,158},{6,32,58,84,110,136,162},{6,26,54,82,110,138,166},{6,30,58,86,114,142,170} -- 35 - 40
}

The alignment pattern has size 5x5 and looks like this:

XXXXX
X   X
X X X
X   X
XXXXX
local function add_alignment_pattern( tab_x )
	local version = (#tab_x - 17) / 4
	local ap = alignment_pattern[version]
	local pos_x, pos_y
	for x=1,#ap do
		for y=1,#ap do
			-- we must not put an alignment pattern on top of the positioning pattern
			if not (x == 1 and y == 1 or x == #ap and y == 1 or x == 1 and y == #ap ) then
				pos_x = ap[x] + 1
				pos_y = ap[y] + 1
				tab_x[pos_x][pos_y] = 2
				tab_x[pos_x+1][pos_y] = -2
				tab_x[pos_x-1][pos_y] = -2
				tab_x[pos_x+2][pos_y] =  2
				tab_x[pos_x-2][pos_y] =  2
				tab_x[pos_x  ][pos_y - 2] = 2
				tab_x[pos_x+1][pos_y - 2] = 2
				tab_x[pos_x-1][pos_y - 2] = 2
				tab_x[pos_x+2][pos_y - 2] = 2
				tab_x[pos_x-2][pos_y - 2] = 2
				tab_x[pos_x  ][pos_y + 2] = 2
				tab_x[pos_x+1][pos_y + 2] = 2
				tab_x[pos_x-1][pos_y + 2] = 2
				tab_x[pos_x+2][pos_y + 2] = 2
				tab_x[pos_x-2][pos_y + 2] = 2

				tab_x[pos_x  ][pos_y - 1] = -2
				tab_x[pos_x+1][pos_y - 1] = -2
				tab_x[pos_x-1][pos_y - 1] = -2
				tab_x[pos_x+2][pos_y - 1] =  2
				tab_x[pos_x-2][pos_y - 1] =  2
				tab_x[pos_x  ][pos_y + 1] = -2
				tab_x[pos_x+1][pos_y + 1] = -2
				tab_x[pos_x-1][pos_y + 1] = -2
				tab_x[pos_x+2][pos_y + 1] =  2
				tab_x[pos_x-2][pos_y + 1] =  2
			end
		end
	end
end

Type information

Let's not forget the type information that is in column 9 next to the left positioning patterns and on row 9 below the top positioning patterns. This type information is not fixed, it depends on the mask and the error correction.

-- The first index is ec level (LMQH,1-4), the second is the mask (0-7). This bitstring of length 15 is to be used
-- as mandatory pattern in the qrcode. Mask -1 is for debugging purpose only and is the 'noop' mask.
local typeinfo = {
	{ [-1]= "111111111111111", [0] = "111011111000100", "111001011110011", "111110110101010", "111100010011101", "110011000101111", "110001100011000", "110110001000001", "110100101110110" },
	{ [-1]= "111111111111111", [0] = "101010000010010", "101000100100101", "101111001111100", "101101101001011", "100010111111001", "100000011001110", "100111110010111", "100101010100000" },
	{ [-1]= "111111111111111", [0] = "011010101011111", "011000001101000", "011111100110001", "011101000000110", "010010010110100", "010000110000011", "010111011011010", "010101111101101" },
	{ [-1]= "111111111111111", [0] = "001011010001001", "001001110111110", "001110011100111", "001100111010000", "000011101100010", "000001001010101", "000110100001100", "000100000111011" }
}

-- The typeinfo is a mixture of mask and ec level information and is
-- added twice to the qr code, one horizontal, one vertical.
local function add_typeinfo_to_matrix( matrix,ec_level,mask )
	local ec_mask_type = typeinfo[ec_level][mask]

	local bit
	-- vertical from bottom to top
	for i=1,7 do
		bit = string.sub(ec_mask_type,i,i)
		fill_matrix_position(matrix, bit, 9, #matrix - i + 1)
	end
	for i=8,9 do
		bit = string.sub(ec_mask_type,i,i)
		fill_matrix_position(matrix,bit,9,17-i)
	end
	for i=10,15 do
		bit = string.sub(ec_mask_type,i,i)
		fill_matrix_position(matrix,bit,9,16 - i)
	end
	-- horizontal, left to right
	for i=1,6 do
		bit = string.sub(ec_mask_type,i,i)
		fill_matrix_position(matrix,bit,i,9)
	end
	bit = string.sub(ec_mask_type,7,7)
	fill_matrix_position(matrix,bit,8,9)
	for i=8,15 do
		bit = string.sub(ec_mask_type,i,i)
		fill_matrix_position(matrix,bit,#matrix - 15 + i,9)
	end
end

-- Bits for version information 7-40
local version_information = {"001010010011111000", "000111101101000100", "100110010101100100","011001011001010100",
  "011011111101110100", "001000110111001100", "111000100001101100", "010110000011011100", "000101001001111100",
  "000111101101000010", "010111010001100010", "111010000101010010", "001001100101110010", "011001011001001010",
  "011000001011101010", "100100110001011010", "000110111111111010", "001000110111000110", "000100001111100110",
  "110101011111010110", "000001110001110110", "010110000011001110", "001111110011101110", "101011101011011110",
  "000000101001111110", "101010111001000001", "000001111011100001", "010111010001010001", "011111001111110001",
  "110100001101001001", "001110100001101001", "001001100101011001", "010000010101111001", "100101100011000101" }

-- Versions 7 and above need two bitfields with version information added to the code
local function add_version_information(matrix,version)
	if version < 7 then return end
	local size = #matrix
	local bitstring = version_information[version - 6]
	local x,y, bit
	local start_x, start_y
	-- first top right
	start_x = #matrix - 10
	start_y = 1
	for i=1,#bitstring do
		bit = string.sub(bitstring,i,i)
		x = start_x + math.fmod(i - 1,3)
		y = start_y + math.floor( (i - 1) / 3 )
		fill_matrix_position(matrix,bit,x,y)
	end

	-- now bottom left
	start_x = 1
	start_y = #matrix - 10
	for i=1,#bitstring do
		bit = string.sub(bitstring,i,i)
		x = start_x + math.floor( (i - 1) / 3 )
		y = start_y + math.fmod(i - 1,3)
		fill_matrix_position(matrix,bit,x,y)
	end
end

Now it's time to use the methods above to create a prefilled matrix for the given mask

local function prepare_matrix_with_mask( version,ec_level, mask )
	local size
	local tab_x = {}

	size = version * 4 + 17
	for i=1,size do
		tab_x[i]={}
		for j=1,size do
			tab_x[i][j] = 0
		end
	end
	add_position_detection_patterns(tab_x)
	add_timing_pattern(tab_x)
	add_version_information(tab_x,version)

	-- black pixel above lower left position detection pattern
	tab_x[9][size - 7] = 2
	add_alignment_pattern(tab_x)
	add_typeinfo_to_matrix(tab_x,ec_level, mask)
	return tab_x
end

Finally we come to the place where we need to put the calculated data (remember step 3?) into the qr code. We do this for each mask. BTW speaking of mask, this is what we find in the spec:

 Mask Pattern Reference   Condition
 000                      (y + x) mod 2 = 0
 001                      y mod 2 = 0
 010                      x mod 3 = 0
 011                      (y + x) mod 3 = 0
 100                      ((y div 2) + (x div 3)) mod 2 = 0
 101                      (y x) mod 2 + (y x) mod 3 = 0
 110                      ((y x) mod 2 + (y x) mod 3) mod 2 = 0
 111                      ((y x) mod 3 + (y+x) mod 2) mod 2 = 0
-- Return 1 (black) or -1 (blank) depending on the mask, value and position.
-- Parameter mask is 0-7 (-1 for 'no mask'). x and y are 1-based coordinates,
-- 1,1 = upper left. tonumber(value) must be 0 or 1.
local function get_pixel_with_mask( mask, x,y,value )
	x = x - 1
	y = y - 1
	local invert = false
	-- test purpose only:
	if mask == -1 then
		-- ignore, no masking applied
	elseif mask == 0 then
		if math.fmod(x + y,2) == 0 then invert = true end
	elseif mask == 1 then
		if math.fmod(y,2) == 0 then invert = true end
	elseif mask == 2 then
		if math.fmod(x,3) == 0 then invert = true end
	elseif mask == 3 then
		if math.fmod(x + y,3) == 0 then invert = true end
	elseif mask == 4 then
		if math.fmod(math.floor(y / 2) + math.floor(x / 3),2) == 0 then invert = true end
	elseif mask == 5 then
		if math.fmod(x * y,2) + math.fmod(x * y,3) == 0 then invert = true end
	elseif mask == 6 then
		if math.fmod(math.fmod(x * y,2) + math.fmod(x * y,3),2) == 0 then invert = true end
	elseif mask == 7 then
		if math.fmod(math.fmod(x * y,3) + math.fmod(x + y,2),2) == 0 then invert = true end
	else
		assert(false,"This can't happen (mask must be <= 7)")
	end
	if invert then
		-- value = 1? -> -1, value = 0? -> 1
		return 1 - 2 * tonumber(value)
	else
		-- value = 1? -> 1, value = 0? -> -1
		return -1 + 2*tonumber(value)
	end
end


-- We need up to 8 positions in the matrix. Only the last few bits may be less then 8.
-- The function returns a table of (up to) 8 entries with subtables where
-- the x coordinate is the first and the y coordinate is the second entry.
local function get_next_free_positions(matrix,x,y,dir,byte)
	local ret = {}
	local count = 1
	local mode = "right"
	while count <= #byte do
		if mode == "right" and matrix[x][y] == 0 then
			ret[#ret + 1] = {x,y}
			mode = "left"
			count = count + 1
		elseif mode == "left" and matrix[x-1][y] == 0 then
			ret[#ret + 1] = {x-1,y}
			mode = "right"
			count = count + 1
			if dir == "up" then
				y = y - 1
			else
				y = y + 1
			end
		elseif mode == "right" and matrix[x-1][y] == 0 then
			ret[#ret + 1] = {x-1,y}
			count = count + 1
			if dir == "up" then
				y = y - 1
			else
				y = y + 1
			end
		else
			if dir == "up" then
				y = y - 1
			else
				y = y + 1
			end
		end
		if y < 1 or y > #matrix then
			x = x - 2
			-- don't overwrite the timing pattern
			if x == 7 then x = 6 end
			if dir == "up" then
				dir = "down"
				y = 1
			else
				dir = "up"
				y = #matrix
			end
		end
	end
	return ret,x,y,dir
end

-- Add the data string (0's and 1's) to the matrix for the given mask.
local function add_data_to_matrix(matrix,data,mask)
	size = #matrix
	local x,y,positions
	local _x,_y,m
	local dir = "up"
	local byte_number = 0
	x,y = size,size
	string.gsub(data,".?.?.?.?.?.?.?.?",function ( byte )
		byte_number = byte_number + 1
		positions,x,y,dir = get_next_free_positions(matrix,x,y,dir,byte,mask)
		for i=1,#byte do
			_x = positions[i][1]
			_y = positions[i][2]
			m = get_pixel_with_mask(mask,_x,_y,string.sub(byte,i,i))
			if debugging then
				matrix[_x][_y] = m * (i + 10)
			else
				matrix[_x][_y] = m
			end
		end
	end)
end

The total penalty of the matrix is the sum of four steps. The following steps are taken into account:

  1. Adjacent modules in row/column in same color
  2. Block of modules in same color
  3. 1:1:3:1:1 ratio (dark:light:dark:light:dark) pattern in row/column
  4. Proportion of dark modules in entire symbol

This all is done to avoid bad patterns in the code that prevent the scanner from reading the code.

-- Return the penalty for the given matrix
local function calculate_penalty(matrix)
	local penalty1, penalty2, penalty3, penalty4 = 0,0,0,0
	local size = #matrix
	-- this is for penalty 4
	local number_of_dark_cells = 0

	-- 1: Adjacent modules in row/column in same color
	-- --------------------------------------------
	-- No. of modules = (5+i)  -> 3 + i
	local last_bit_blank -- < 0:  blank, > 0: black
	local is_blank
	local number_of_consecutive_bits
	-- first: vertical
	for x=1,size do
		number_of_consecutive_bits = 0
		last_bit_blank = nil
		for y = 1,size do
			if matrix[x][y] > 0 then
				-- small optimization: this is for penalty 4
				number_of_dark_cells = number_of_dark_cells + 1
				is_blank = false
			else
				is_blank = true
			end
			is_blank = matrix[x][y] < 0
			if last_bit_blank == is_blank then
				number_of_consecutive_bits = number_of_consecutive_bits + 1
			else
				if number_of_consecutive_bits >= 5 then
					penalty1 = penalty1 + number_of_consecutive_bits - 2
				end
				number_of_consecutive_bits = 1
			end
			last_bit_blank = is_blank
		end
		if number_of_consecutive_bits >= 5 then
			penalty1 = penalty1 + number_of_consecutive_bits - 2
		end
	end
	-- now horizontal
	for y=1,size do
		number_of_consecutive_bits = 0
		last_bit_blank = nil
		for x = 1,size do
			is_blank = matrix[x][y] < 0
			if last_bit_blank == is_blank then
				number_of_consecutive_bits = number_of_consecutive_bits + 1
			else
				if number_of_consecutive_bits >= 5 then
					penalty1 = penalty1 + number_of_consecutive_bits - 2
				end
				number_of_consecutive_bits = 1
			end
			last_bit_blank = is_blank
		end
		if number_of_consecutive_bits >= 5 then
			penalty1 = penalty1 + number_of_consecutive_bits - 2
		end
	end
	for x=1,size do
		for y=1,size do
			-- 2: Block of modules in same color
			-- -----------------------------------
			-- Blocksize = m × n  -> 3 × (m-1) × (n-1)
			if (y < size - 1) and ( x < size - 1) and ( (matrix[x][y] < 0 and matrix[x+1][y] < 0 and matrix[x][y+1] < 0 and matrix[x+1][y+1] < 0) or (matrix[x][y] > 0 and matrix[x+1][y] > 0 and matrix[x][y+1] > 0 and matrix[x+1][y+1] > 0) ) then
				penalty2 = penalty2 + 3
			end

			-- 3: 1:1:3:1:1 ratio (dark:light:dark:light:dark) pattern in row/column
			-- ------------------------------------------------------------------
			-- Gives 40 points each
			--
			-- I have no idea why we need the extra 0000 on left or right side. The spec doesn't mention it,
			-- other sources do mention it. This is heavily inspired by zxing.
			if (y + 6 < size and
				matrix[x][y] > 0 and
				matrix[x][y +  1] < 0 and
				matrix[x][y +  2] > 0 and
				matrix[x][y +  3] > 0 and
				matrix[x][y +  4] > 0 and
				matrix[x][y +  5] < 0 and
				matrix[x][y +  6] > 0 and
				((y + 10 < size and
					matrix[x][y +  7] < 0 and
					matrix[x][y +  8] < 0 and
					matrix[x][y +  9] < 0 and
					matrix[x][y + 10] < 0) or
				 (y - 4 >= 1 and
					matrix[x][y -  1] < 0 and
					matrix[x][y -  2] < 0 and
					matrix[x][y -  3] < 0 and
					matrix[x][y -  4] < 0))) then penalty3 = penalty3 + 40 end
			if (x + 6 <= size and
				matrix[x][y] > 0 and
				matrix[x +  1][y] < 0 and
				matrix[x +  2][y] > 0 and
				matrix[x +  3][y] > 0 and
				matrix[x +  4][y] > 0 and
				matrix[x +  5][y] < 0 and
				matrix[x +  6][y] > 0 and
				((x + 10 <= size and
					matrix[x +  7][y] < 0 and
					matrix[x +  8][y] < 0 and
					matrix[x +  9][y] < 0 and
					matrix[x + 10][y] < 0) or
				 (x - 4 >= 1 and
					matrix[x -  1][y] < 0 and
					matrix[x -  2][y] < 0 and
					matrix[x -  3][y] < 0 and
					matrix[x -  4][y] < 0))) then penalty3 = penalty3 + 40 end
		end
	end
	-- 4: Proportion of dark modules in entire symbol
	-- ----------------------------------------------
	-- 50 ± (5 × k)% to 50 ± (5 × (k + 1))% -> 10 × k
	local dark_ratio = number_of_dark_cells / ( size * size )
	penalty4 = math.floor(math.abs(dark_ratio * 100 - 50)) * 2
	return penalty1 + penalty2 + penalty3 + penalty4
end

-- Create a matrix for the given parameters and calculate the penalty score.
-- Return both (matrix and penalty)
local function get_matrix_and_penalty(version,ec_level,data,mask)
	local tab = prepare_matrix_with_mask(version,ec_level,mask)
	add_data_to_matrix(tab,data,mask)
	local penalty = calculate_penalty(tab)
	return tab, penalty
end

-- Return the matrix with the smallest penalty. To to this
-- we try out the matrix for all 8 masks and determine the
-- penalty (score) each.
local function get_matrix_with_lowest_penalty(version,ec_level,data)
	local tab, penalty
	local tab_min_penalty, min_penalty

	-- try masks 0-7
	tab_min_penalty, min_penalty = get_matrix_and_penalty(version,ec_level,data,0)
	for i=1,7 do
		tab, penalty = get_matrix_and_penalty(version,ec_level,data,i)
		if penalty < min_penalty then
			tab_min_penalty = tab
			min_penalty = penalty
		end
	end
	return tab_min_penalty
end

The main function. We connect everything together. Remember from above:

  1. Determine version, ec level and mode (=encoding) for codeword
  2. Encode data
  3. Arrange data and calculate error correction code
  4. Generate 8 matrices with different masks and calculate the penalty
  5. Return qrcode with least penalty
-- If ec_level or mode is given, use the ones for generating the qrcode. (mode is not implemented yet)
local function qrcode( str, ec_level, mode )
	local arranged_data, version, data_raw, mode, len_bitstring
	version, ec_level, data_raw, mode, len_bitstring = get_version_eclevel_mode_bistringlength(str,ec_level)
	data_raw = data_raw .. len_bitstring
	data_raw = data_raw .. encode_data(str,mode)
	data_raw = add_pad_data(version,ec_level,data_raw)
	arranged_data = arrange_codewords_and_calculate_ec(version,ec_level,data_raw)
	if math.fmod(#arranged_data,8) ~= 0 then
		return false, string.format("Arranged data %% 8 != 0: data length = %d, mod 8 = %d",#arranged_data, math.fmod(#arranged_data,8))
	end
	arranged_data = arranged_data .. string.rep("0",remainder[version])
	local tab = get_matrix_with_lowest_penalty(version,ec_level,arranged_data)
	return true, tab
end


if testing then
	return {
		encode_string_numeric = encode_string_numeric,
		encode_string_ascii = encode_string_ascii,
		qrcode = qrcode,
		binary = binary,
		get_mode = get_mode,
		get_length = get_length,
		add_pad_data = add_pad_data,
		get_generator_polynominal_adjusted = get_generator_polynominal_adjusted,
		get_pixel_with_mask = get_pixel_with_mask,
		get_version_eclevel_mode_bistringlength = get_version_eclevel_mode_bistringlength,
		remainder = remainder,
		--get_capacity_remainder = get_capacity_remainder,
		arrange_codewords_and_calculate_ec = arrange_codewords_and_calculate_ec,
		calculate_error_correction = calculate_error_correction,
		convert_bitstring_to_bytes = convert_bitstring_to_bytes,
		bit_xor = bit_xor,
	}
end

return {
	qrcode = qrcode
}