# Parity Blocks

- **Track:** Cambridge IGCSE™ Computer Science (0478 / 0984) (CAMBRIDGE IGCSE 0478)
- **Module:** Section 2: Data Transmission
- **Estimated Duration:** 15 minutes
- **Access:** Open Theory (All-Access for Interactive IDE & Autotests)

---

## Theory & Conceptual Background

## 2D Parity Blocks (Matrix Parity)

A **2D Parity Block** (also called horizontal and vertical parity) can locate and correct an assumed single-bit error and detect many, but not all, multiple-bit errors.

### How a Parity Block Works:

1. Data is transmitted as a block of multiple bytes arranged in a table (grid).
2. **Horizontal Parity**: Each individual byte (row) contains its own parity bit.
3. **Vertical Parity**: An entire extra byte (the **Parity Byte**) is calculated and appended at the bottom, containing a parity bit for each column.

### Worked Example: Even Parity Block

Suppose 4 bytes of data are transmitted using **Even Parity**:

```text
Col 7  Col 6  Col 5  Col 4  Col 3  Col 2  Col 1  [Parity Bit]   Row Parity Status
Byte 1:        1      0      1      1      0      0      1          0        -> 4 ones (Valid Even)
Byte 2:        0      1      1      0      1      1      0          0        -> 4 ones (Valid Even)
Byte 3:        1      1      0      1      0      0      0          1        -> 4 ones (Valid Even)
Byte 4:        0      0      1      1      1      0      1          0        -> 4 ones (Valid Even)
-------------------------------------------------------------------------
Parity Byte:   0      0      1      1      0      1      0          1        -> Column Parity
```

### Detecting and Correcting a Flipped Bit

Suppose during transmission, electrical noise flips **Row 2, Column 5** from `1` to `0`:

```text
Col 7  Col 6  Col 5  Col 4  Col 3  Col 2  Col 1  [Parity Bit]
Byte 1:        1      0      1      1      0      0      1          0       -> OK
Byte 2:        0      1     [0]     0      1      1      0          0       -> ERROR! (3 ones)
Byte 3:        1      1      0      1      0      0      0          1       -> OK
Byte 4:        0      0      1      1      1      0      1          0       -> OK
-------------------------------------------------------------------------
Parity Byte:   0      0      1      1      0      1      0          1
               OK     OK   ERROR!   OK     OK     OK     OK        OK
```

1. **Row Check**: Row 2 now has three 1s instead of an even number $\implies$ **Row 2 has an error**.
2. **Column Check**: Column 5 now has three 1s instead of four, including its parity bit $\implies$ **Column 5 has an error**.
3. **Intersection**: The intersection of **Row 2** and **Column 5** identifies the exact erroneous bit.
4. **Correction**: The receiver inverts the bit from `0` back to `1`. The data is restored to pristine accuracy without requesting a slow network retransmission!

## Limits and independent practice

Always count the parity bit when checking a row or column. An odd result where even parity is required indicates a mismatch. With an assumed single-bit error, the failed row and column identify its location. If four corners of a rectangle flip, each affected row and column changes twice and the errors can remain undetected. Do not claim arbitrary multi-bit correction.

---

## Interactive Exercises & Autotest Specifications

### Exercise 1: Application: Locate a bit error (Fill in blank, 30 XP)

**Instructions:**
Work out each answer before submitting. An even-parity block has only row 3 and column 2 failing. Which row contains the single-bit error? Which column? If all four corners of a rectangle flip, can every parity check still pass: yes or no?


**Starter Code:**
```text
{"prompt":"Application: Locate a bit error","template":"1. An even-parity block has only row 3 and column 2 failing. Which row contains the single-bit error? [blank_1]\n2. Which column? [blank_2]\n3. If all four corners of a rectangle flip, can every parity check still pass: yes or no? [blank_3]","blanks":[{"id":"blank_1"},{"id":"blank_2"},{"id":"blank_3"}],"word_bank":["3","2","yes","1","4","no"]}
```


### Exercise 2: Independent Practice: Check an entire block (Fill in blank, 30 XP)

**Instructions:**
Work out each answer before submitting. Rows including even parity are 1010, 1101, 0110; the final parity row is 0000. Number of 1s in row 2: Number of 1s in column 4 including its parity bit: Assuming one error, correct row 2 as four binary digits:


**Starter Code:**
```text
{"prompt":"Independent Practice: Check an entire block","template":"1. Rows including even parity are 1010, 1101, 0110; the final parity row is 0000. Number of 1s in row 2: [blank_1]\n2. Number of 1s in column 4 including its parity bit: [blank_2]\n3. Assuming one error, correct row 2 as four binary digits: [blank_3]","blanks":[{"id":"blank_1"},{"id":"blank_2"},{"id":"blank_3"}],"word_bank":["3","1","1100","2","4","0","1101","1001"]}
```