Maths and Computing:Encode Version 1 QR Code

QR Code Masks

QRCode Masks

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

Mask Pattern 0

(row + col) % 2 == 0

Mask Pattern 1

row % 2 == 0

Mask Pattern 2

col % 3 == 0

Mask Pattern 3

(row + col) % 3 == 0

Mask Pattern 4

floor(row/2) + floor(col/3)) % 2 == 0

Mask Pattern 5

(row * col) % 2 + (row * col) % 3 == 0

Mask Pattern 6

((row * col) % 2 + (row * col) % 3) % 2 == 0

Mask Pattern 7

((row * col) % 3 + row + col) % 2 == 0

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

Mappings from the original colour to the new colour depending on the result of the conditional statement

Original Module Color	Result of Condition Statement	New Module Color
white	false	white
white	true	black
black	false	black
black	true	white

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.

Output from the XOR operation and the equivalent in terms of the module colour

A	B	A XOR B	Module Equivalent
0	0	0	W	false	W
0	1	1	W	true	B
1	0	1	B	false	B
1	1	0	B	true	W

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%

Mask Pattern 0

(row + col) % 2 == 0

=>

Penalty Points 1: 181

Penalty Points 2: 111

Penalty Points 3: 80

Penalty Points 4: 0

Total: 372

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

12 runs of white horizontal blocks

9 additional blocks

Penalty: 12 × 3 + 9 = 45

8 runs of black horizontal blocks

16 additional blocks

Penalty: 8 × 3 + 16 = 40

13 runs of white vertical blocks

20 additional blocks

Penalty: 13 × 3 + 20 = 59

8 runs of black vertical blocks

13 additional blocks

Penalty: 8 × 3 + 13 = 37

Total penalty points: 181

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

16 white 2 by 2 blocks

Penalty: 16 × 3 = 48

21 black 2 by 2 blocks

Penalty: 21 × 3 = 63

Total penalty points: 37 × 3 = 111

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

0 horizontal matching runs

2 vertical matching runs

Total penalty points: 80

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

215 white pixels

226 black pixels

Percentage black pixels: (226 × 100) ÷ (21 × 21) = 51%

(51 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 1

row % 2 == 0

=>

Penalty Points 1: 197

Penalty Points 2: 144

Penalty Points 3: 160

Penalty Points 4: 0

Total: 501

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

11 runs of white horizontal blocks

11 additional blocks

Penalty: 11 × 3 + 11 = 44

9 runs of black horizontal blocks

28 additional blocks

Penalty: 9 × 3 + 28 = 55

11 runs of white vertical blocks

13 additional blocks

Penalty: 11 × 3 + 13 = 46

10 runs of black vertical blocks

22 additional blocks

Penalty: 10 × 3 + 22 = 52

Total penalty points: 197

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

14 white 2 by 2 blocks

Penalty: 14 × 3 = 42

34 black 2 by 2 blocks

Penalty: 34 × 3 = 102

Total penalty points: 48 × 3 = 144

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

3 horizontal matching runs

1 vertical matching runs

Total penalty points: 160

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

203 white pixels

238 black pixels

Percentage black pixels: (238 × 100) ÷ (21 × 21) = 53%

(53 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 2

col % 3 == 0

=>

Penalty Points 1: 185

Penalty Points 2: 108

Penalty Points 3: 80

Penalty Points 4: 0

Total: 373

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

10 runs of white horizontal blocks

13 additional blocks

Penalty: 10 × 3 + 13 = 43

10 runs of black horizontal blocks

22 additional blocks

Penalty: 10 × 3 + 22 = 52

11 runs of white vertical blocks

9 additional blocks

Penalty: 11 × 3 + 9 = 42

12 runs of black vertical blocks

12 additional blocks

Penalty: 12 × 3 + 12 = 48

Total penalty points: 185

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

9 white 2 by 2 blocks

Penalty: 9 × 3 = 27

27 black 2 by 2 blocks

Penalty: 27 × 3 = 81

Total penalty points: 36 × 3 = 108

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

1 horizontal matching runs

1 vertical matching runs

Total penalty points: 80

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

205 white pixels

236 black pixels

Percentage black pixels: (236 × 100) ÷ (21 × 21) = 53%

(53 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 3

(row + col) % 3 == 0

=>

Penalty Points 1: 174

Penalty Points 2: 123

Penalty Points 3: 0

Penalty Points 4: 0

Total: 297

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

11 runs of white horizontal blocks

12 additional blocks

Penalty: 11 × 3 + 12 = 45

9 runs of black horizontal blocks

22 additional blocks

Penalty: 9 × 3 + 22 = 49

9 runs of white vertical blocks

10 additional blocks

Penalty: 9 × 3 + 10 = 37

10 runs of black vertical blocks

13 additional blocks

Penalty: 10 × 3 + 13 = 43

Total penalty points: 174

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

11 white 2 by 2 blocks

Penalty: 11 × 3 = 33

30 black 2 by 2 blocks

Penalty: 30 × 3 = 90

Total penalty points: 41 × 3 = 123

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

0 horizontal matching runs

0 vertical matching runs

Total penalty points: 0

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

207 white pixels

234 black pixels

Percentage black pixels: (234 × 100) ÷ (21 × 21) = 53%

(53 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 4

floor(row/2) + floor(col/3)) % 2 == 0

=>

Penalty Points 1: 198

Penalty Points 2: 135

Penalty Points 3: 160

Penalty Points 4: 0

Total: 493

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

16 runs of white horizontal blocks

14 additional blocks

Penalty: 16 × 3 + 14 = 62

8 runs of black horizontal blocks

13 additional blocks

Penalty: 8 × 3 + 13 = 37

11 runs of white vertical blocks

22 additional blocks

Penalty: 11 × 3 + 22 = 55

9 runs of black vertical blocks

17 additional blocks

Penalty: 9 × 3 + 17 = 44

Total penalty points: 198

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

22 white 2 by 2 blocks

Penalty: 22 × 3 = 66

23 black 2 by 2 blocks

Penalty: 23 × 3 = 69

Total penalty points: 45 × 3 = 135

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

2 horizontal matching runs

2 vertical matching runs

Total penalty points: 160

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

215 white pixels

226 black pixels

Percentage black pixels: (226 × 100) ÷ (21 × 21) = 51%

(51 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 5

(row * col) % 2 + (row * col) % 3 == 0

=>

Penalty Points 1: 180

Penalty Points 2: 159

Penalty Points 3: 80

Penalty Points 4: 0

Total: 419

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

9 runs of white horizontal blocks

19 additional blocks

Penalty: 9 × 3 + 19 = 46

7 runs of black horizontal blocks

19 additional blocks

Penalty: 7 × 3 + 19 = 40

11 runs of white vertical blocks

15 additional blocks

Penalty: 11 × 3 + 15 = 48

11 runs of black vertical blocks

13 additional blocks

Penalty: 11 × 3 + 13 = 46

Total penalty points: 180

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

25 white 2 by 2 blocks

Penalty: 25 × 3 = 75

28 black 2 by 2 blocks

Penalty: 28 × 3 = 84

Total penalty points: 53 × 3 = 159

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

0 horizontal matching runs

2 vertical matching runs

Total penalty points: 80

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

215 white pixels

226 black pixels

Percentage black pixels: (226 × 100) ÷ (21 × 21) = 51%

(51 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 6

((row * col) % 2 + (row * col) % 3) % 2 == 0

=>

Penalty Points 1: 191

Penalty Points 2: 156

Penalty Points 3: 80

Penalty Points 4: 0

Total: 427

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

12 runs of white horizontal blocks

15 additional blocks

Penalty: 12 × 3 + 15 = 51

10 runs of black horizontal blocks

23 additional blocks

Penalty: 10 × 3 + 23 = 53

11 runs of white vertical blocks

12 additional blocks

Penalty: 11 × 3 + 12 = 45

10 runs of black vertical blocks

12 additional blocks

Penalty: 10 × 3 + 12 = 42

Total penalty points: 191

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

22 white 2 by 2 blocks

Penalty: 22 × 3 = 66

30 black 2 by 2 blocks

Penalty: 30 × 3 = 90

Total penalty points: 52 × 3 = 156

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

1 horizontal matching runs

1 vertical matching runs

Total penalty points: 80

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

211 white pixels

230 black pixels

Percentage black pixels: (230 × 100) ÷ (21 × 21) = 52%

(52 − 50) ÷ 5) = 0

penalty = 0 * 10 = 0

Total penalty points: 0

Mask Pattern 7

((row * col) % 3 + row + col) % 2 == 0

=>

Penalty Points 1: 168

Penalty Points 2: 90

Penalty Points 3: 120

Penalty Points 4: 0

Total: 378

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

12 runs of white horizontal blocks

11 additional blocks

Penalty: 12 × 3 + 11 = 47

9 runs of black horizontal blocks

14 additional blocks

Penalty: 9 × 3 + 14 = 41

11 runs of white vertical blocks

14 additional blocks

Penalty: 11 × 3 + 14 = 47

7 runs of black vertical blocks

12 additional blocks

Penalty: 7 × 3 + 12 = 33

Total penalty points: 168

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points>

12 white 2 by 2 blocks

Penalty: 12 × 3 = 36

18 black 2 by 2 blocks

Penalty: 18 × 3 = 54

Total penalty points: 30 × 3 = 90

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

0 horizontal matching runs

3 vertical matching runs

Total penalty points: 120

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

Maths and Computing:

QR Code Masks

QRCode Masks

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

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

Mask Pattern 0

(row + col) % 2 == 0

Mask Pattern 1

row % 2 == 0

Mask Pattern 2

col % 3 == 0

Mask Pattern 3

(row + col) % 3 == 0

Mask Pattern 4

floor(row/2) + floor(col/3)) % 2 == 0

Mask Pattern 5

(row * col) % 2 + (row * col) % 3 == 0

Mask Pattern 6

((row * col) % 2 + (row * col) % 3) % 2 == 0

Mask Pattern 7

((row * col) % 3 + row + col) % 2 == 0

Applying a Mask Pattern to a QR Code

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

Understanding the Mask Pattern

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

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.

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.

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%

Mask Pattern 0

(row + col) % 2 == 0

=>

Penalty Points 1: 181

Penalty Points 2: 111

Penalty Points 3: 80

Penalty Points 4: 0

Total: 372

Penalty Calculation 1: Runs of same colour pixels

Each run of 5 or more pixels of the same colour has a penalty of 3 points plus 1 for each additional pixel

12 runs of white horizontal blocks

9 additional blocks

Penalty: 12 × 3 + 9 = 45

8 runs of black horizontal blocks

16 additional blocks

Penalty: 8 × 3 + 16 = 40

13 runs of white vertical blocks

20 additional blocks

Penalty: 13 × 3 + 20 = 59

8 runs of black vertical blocks

13 additional blocks

Penalty: 8 × 3 + 13 = 37

Total penalty points: 181

Penalty Calculation 2: Blocks of the same colour

Each 2 by 2 block of the same colour has a penalty of 3 points> 16 white 2 by 2 blocks

16 white 2 by 2 blocks

Penalty: 16 × 3 = 48

21 black 2 by 2 blocks

Penalty: 21 × 3 = 63

Total penalty points: 37 × 3 = 111

Penalty Calculation 3: Runs matching 10111010000

Each run matching the following has a penalty of 40 points

0 horizontal matching runs

2 vertical matching runs

Total penalty points: 80

Penalty Calculation 4

Penalty calculated from the percentage of black pixels. 10 penality points for each multiple of 5% from 50%

215 white pixels

226 black pixels

Percentage black pixels: (226 × 100) ÷ (21 × 21) = 51%

Each 2 by 2 block of the same colour has a penalty of 3 points>

16 white 2 by 2 blocks

Each 2 by 2 block of the same colour has a penalty of 3 points>

14 white 2 by 2 blocks