Maths and Computing:

Creating a QR Code

This tool is designed to create a version 1 QR code using one of three different options.

Single Step

The QR code is created in one step using optimal settings for determining the encoding, error level and mask value

Step by Step

The settings are entered one step at a time with the available options limited by the results of the previous steps

Custom Entry

Allows the setting for the encoding, error level and mask values to be entered together.

Explanation of the Process

These pages describe the detail relating to creating a version 1 QR code and does this using the example of the QR code that is being created.

Once the QR code has been created, the following information is provided about each step of the process and the different elements involved in creating a QR code.

• The encoding of each character
• The calculation used to create the error codes
• The data placement within the QR code
• The application of the mask pattern on the data
• The creation of the information bits

Single Step

Create the QR code in one step using the optimal encoding, error level values and mask.

The maximum length of text allowed for a version 1 QR code is:

• 41 numeric characters
• 25 alphanumeric character
• 17 binary characters.

If the text entered is too long for the encoding, the text will be truncated to allow it to fit.

Enter the text that you want to create a version 1 QR code for.

Step by Step

Create the QR code step by step, by providing the values of the following:

• The Encoding
• The Error Level
• The Mask Pattern

The available options are based on the output from the previous steps.

Enter the text that you want to create a version 1 QR code for.

Customise

Create the QR code in one step using customised settings for the encoding, error level values and mask

Enter the text that you want to create a version 1 QR code for.

The Key Stages in Creating a QR Code

This is a step by step overview of the processes involved in creating a QR code containing a given series of characters.

For more information, select the title button for a given stage.

• An encoding mode is chosen to determine how the characters will be encoded.
• Binary data can contain 7-bit ASCII characters or unicode characters that are encoded as multibyte characters using the UTF-8 encoding.
• Numeric data takes the least space, followed by alphanumeric and then binary
• If the data can be encoded in more that one way, then the one that takes the least space is chosen

• An error level determines the number of bytes that can be corrupted, detected and correntec.
• The four error levels are high, quartile, medium and low.
• The high error level requires the most space followed by quartile, medium and then low.
• The error level is determined accoring to whether it is more important to be able to correct data or to keep the size of the QR Code low.

• With the input data, the encoding and the error level, the number of data bytes can be calculated..
• The number of data bytes determines the minimum QR Code version that can be used..
• The QR Code grid can then be set up with the number of module based on the verison 1
• The static elements of the QR Code are the positioning patterns, the timing patterns and the spacing. These can be added to the grid.

• There are eight different mask patterns.
• Any of these mask patterns can be applied to the data to create a valid QR Code.
• Certain mask patterns create a better result with fewer blocks of white and black data.
• The result when each pattern is applied is rated, and the mask with the lowest ranking is chosen and applied to the data.

• The format information consists of fifteen bits.
• Two bits are used for the error level and three for the mask pattern. The rest are error correction bits.
• A mask is applied to the bits and the resulting bits are added to the QR Code.
• The information bits are placed near to the two left positioning pattern
• A copy of the fifteen bits is also added to the QR Code.

Terminology

Version 1 QR Codes

Detail relating to a Version 1 QR Code

• The QR Code is made up of 21 by 21 black and white modules
• The QR Code has a maximum capacity of 26 bytes of data.
• The maximum number of ASCII characters that can be encoded is seventeen

This tool is designed for a version 1 QR Code and followiing modifications are implement when creating the QR Code.

• If the size of the data is too large for a version 1 QR Code, it will be truncated to fit in
• The available error levels is restricted by the number of bytes left after the data has been encoded

QR Code Encoding

Encoding describes the method by which the characters are converted into bits of data that can be store in the QR code

• There are four standard encoding modes used by QR codes: numeric, alphanumeric, binary and kanji.
• This tool demonstrates numeric, alphanumeric and binary encoding.
• All characters can be encoded using binary encoding although this may not be the most efficient selection.
• The selected encoding is stored as a 4-bit mode indicator at the beginning of the data.

Table of encoding types for a QR code

Numeric Encoding

Numeric encoding can be used if the data is made up entirely of the characters 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9

Encoding the Data

To encode the numeric characters:

• The digits are placed into groups of three from left to right to create values between 0 and 999.
• The 3-digit values are coverted to 10-bit binary numbers.
• If there is one remaining digit, it is multiplied by 16 and converted to an 8-bit binary number
• If there are two remaining digits, 2-digit value is multiplied by 2. The result is converted to an 8-bit binary number

Example: 12345

Digit groups: 123 45

Calculation for 123

• 123 = 0001111011 (10-bit binary value)

Calculation for 45

• 45 × 2 = 90
• 90 = 01011010 (8-bit binary value)

Example: 9876543

Digit groups: 987, 654 and 3

Calculation for 987

• 987 = 1111011011 (10-bit binary value)

Calculation for 654

• 654 = 1010001110 (10-bit binary value)

Calculation for 3

• 3 × 16 = 48
• 48 = 00110000 (8-bit binary value)

Alphanumeric Encoding

Alphanumeric encoding can be used if the data is made up entirely of the following:

• Numeric digits
• Upper case letters
• Space
• The characters $ % * + - , / .

Encoding the Data

To encode the alphanumeric characters:

• Look up the value of each alphanumeric character.
• The values are taken in pairs.
• The first value of each pair is multiplied by 45 and added to the second value to form a value between 1 and 2024.
• This value is then converted to an 11-bit binary number.
• If there is one remaining character, its value is multiplied by 4 and converted to an 8-bit binary number.

Example: CAT

Character groups: CA T

Calculation for C and A

• The values for C and A are 12 and 10
• 45 × 12 + 10 = 550
• 550 = 01000100110 (11-bit binary value)

Calculation for T

• The value for T is 29
• 4 × 29 = 29
• 29 = 01110100 (8-bit binary value)

Binary Encoding

Binary encoding can be used for any character that has a unicode value and is stored using one of more bytes using UTF-8 encoding.

For values between 0 and 127, one byte of data is used and the character is encoded using it's ASCII value.

Multi-Byte Characters

For character with unicode values greater than 127, UTF-8 encoding is used and each character is encoded using two or more bytes of data.

The table below shows examples of such characters and the UTF-8 encoded byte values store in the QR code

Encoding a character as UTF-8

To encode a unicode character value as UTF-8:

• Convert the unicode value to binary adjusting the preceding 0s so that the number starts with 01.
• Split the binary values into groups of six bits from right to left.
• Count the number of bits in the first group and add this to the number of groups/
• If the total is greater than 8, add one to the group count and add an extra group of value 0/
• Create an 8-bit binary value starting with a 1 for each group and ending in os. Add this to the first value.
• Add 10000000 to the value of each remaining group/

The result is to create a multibyte character with the first byte representing the number of bytes, one for each 1 at the beginning, and the other bytes starting with 10 to indicate a continuation bytes.

Example: π

The unicode value for π is 960

• 960 is represented by the binary sequence 011 1100 0000
• Split into groups: 01111 000000
• 2 groups plus 5 initial bits = 7, so no extra groups need to be added
• 11000000 + 01111 = 11001111
• The 8-bit binary value 11001111 = 207
• 10000000 + 000000 = 10000000
• The 8-bit binary value 10000000 = 128

The UTF-8 Encoding for π is 207 128

Example: 🐼

The unicode value for 🐼 is 128060

• 128060 is represented by the binary sequence 01 1111 0100 0011 1100
• Split into groups: 011111 010000 111100
• 3 groups plus 7 initial bits = 10, so as extra groups needs to be added
• 11110000 + 0 = 11110000
• The 8-bit binary value 11110000 = 240
• 10000000 + 011111 = 10011111
• The 8-bit binary value 10011111 = 159
• 10000000 + 010000 = 10010000
• The 8-bit binary value 10010000 = 144
• 10000000 + 111100 = 10111100
• The 8-bit binary value 10111100 = 188

The UTF-8 Encoding for 🐼 is 240 159 144 188

Error Levels

QR codes may be corrupted through incorrect formatting, damage or issues with scanning. The error level determines the percentage of the data that can be corrupted and corrected in the QR Code data.

• There are four error levels: High, Quartile, Medium and Low.
• The error level value is stored in the first two digits of the information string.
• A higher error level is able to correct more data errors but requires more data bytes to be used as error correction codes.
• A lower level requires fewer data bytes to be used as error codes but and correct fewer errors.

Error Levels for a Version 1 QR Code

The table below gives information relating to the maximum number of characters that can be encoded for a particular error level and the maximum percentage of the data that can be corrected for a Version 1 QR code.

Padding Blocks

The number of bytes allocated to the encoded data depends on the error level. If the space used to store the indicator modes and the encoded characters is less that the that available, extra padding blocks are added to to fill up the data.

• The first padding block is made up of four 0 bits and is always included to indicate the end of the data blocks..
• Additional padding blocks standardly use the byte values 0xEC and 0x11, alternating, until less than eight bits of data are left.
• Any final padding block with less that eight bits can take a value of 0

Example: "Version 1" encoded using binary mode

• A version 1 QR code has capacity for 26 bytes.
• The indicator data and extra 0 bits take up two bytes.
• Binary encoded "Version 1" takes up nine bytes (One byte per character).
• This leaves fifteen bytes for the error codes and the padding bytes

Error Codes

To allow for a level of error detection and correctiong, Reed-Solomon error correction codes are added to the data

• The error level determines the number of error correction codes that can be created.
• The current data is split into byte size values, and these are used to generate the error codes
• If the data is corrupted, an algorithm is run on all of the bytes to determinine where the errors occured and how to correct it
• The error level determines at what point there is too much corruption of the data for the damage to be repaired

Generating the Reed-Solomon Error Codes

Generating the Reed-Soloman error codes from the existing data requires the maths of Galois Fields and Polynomial Division.

• The data is split into k bytes
• If the capacity of the QR code is n bytes, then n-k bytes are used as error codes
• The data bytes are used as the coefficients of a polynomial or order k
• The polynomial is then multiplied by x^n-k and divided by a given generator polynomial using Galois Field multiplication over eight elements.
• The coefficients of the remainder are used as the error codes

The "Error Codes"/"Calculation" tab shows the full calculation for any data input when a QR code is generated.

Masking

A mask is applied to the data area of the QR Code to optimise the ability for a scanner to read it.

• There are eight different mask patterns.
• All eight masks produce valid QR codes that can be scanned in.
• QR code generation tools select the one with the lowest penalty as that is considered visibly the best.
• The best masks optimise the distribution of black and white modules.

Mask Patterns

The list of mask patterns show the conditional statement and the result of applying it to a 24 by 24 area of white modules

Applying a Mask Pattern to a QR Code

Mask patterns are applied to the data within a QR code using the conditional statement.

• The row value of a module is the number of rows down from the top of the QR code, starting at 0.
• The column value of a module is the number of columns across from the left of the QR code, starting at 0.
• The row and column values are used to determine if the result of the condition is true or false.
• If the result is true, then the module value changes from black to white or from white ro black
• If the result is false, then the module value stays the same

Understanding the Mask Pattern

Each mask pattern is defined using a conditional statement that uses the modulo (%) operator.

• The modulo operation a % b finds the remainder when a is divided by b.
• By using %2, the result is guarenteed to be always 1 or .0
• For each mask a calculation is done using the row and/or column values of each module and then the modulo 2 operator is applied.
• If the result is 0, then the output is 1.
• If the result is 1, then the output is 0.
• The XOR operation is then applied between the output from the condition and the module value to deternime the updated value of the module.

Using the XOR operator

When applying a mask to a qrcode, the black modules are represented by the value 1 and the white by the value 0.

• If condition is true, then the value is 1.
• If condition is false then the value is 0.
• The XOR operation is then applied between the result of the condition and the value of the module.

Rating Masks

To determine which is the best mask to use for a QR code, each mask is applied and then rated using a penalty point system.

• The mask with the lowest penalty points is chosen by default.
• Any mask value can be chosen although it may not be visibly the best.

Penalty Points

Penalty points are determined using the following factors

Calculation 1: Horizontal and vertical runs of five or more modules of the same colour

Each run of five or more modules of the same colour has a penalty of 3 points plus 1 for each additional pixel

Calculation 2: Block of the same colour greater than two by two modules.

Each two by two block of the same colour has a penalty of 3 points.

Calculation 3: Hoizontal and vertical runs matching the value "10111010000".

Each run matching the following has a penalty of 40 points.

Calculation 4: The ratio of black to white modules

Penalty calculated from the percentage of black modules. Ten penality points for each multiple of 5% from 50%

Components of a QR Code

A QR code consists of a number of static elements that help determine the orientation and size of a QR code when it is being scanned.

The rest of the QR code is used to store the data relating to the information that is encoded within it.

• This QR code structure is for a version 1 QR code.
• The information in the QR code relates to Hello World with Binary encoding, Low error level and mask pattern 1.

Data Position in the QR Code

A version 1 QR code allows for 208 bits of data. This data is entered into the data area of the QR code

• The first bit is placed in the bottom right of the QR code.
• The last bit is placed on the far right just above the bottom left positioning pattern.
• The data goes up and down in a two module width columns, as shown below.
• If the data sequence reaches a point where it cannot continue, it jumps to the bottom right of the unused data space.

Table showing how data is positioned within the QR code

Data bits

The encoded data is then made up of:

1. The mode indicator for the encoding (4 bits)
2. The character count indicator block (The number of bits used depends on the encoding and the QR version)
3. The encoded data blocks
4. Padding data to complete any unused bytes.
5. Error Codes

Format Information

The format information is a 15-bit sequence containing the error level and mask pattern.

• The essential information is made up of five bits.
• The first two bits represent the error level.
• The next three bits represent the mask pattern.

Example: Low Error Level and Mask Pattern 0

Mapping for each Error Level and Mask Combination