# Q7(b) · Block coding

Subject: Computer Networks

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/cn/#tee/papers/paper-2025-legacy-reexam/q7b

Canonical: https://deep-learning-study-room.vercel.app/read/cn/tee/papers/paper-2025-legacy-reexam--q7b/

Actual paper; 100 marks attempted. Current-syllabus alignment is stated per question.

### Q7(b). Explain block codes in detail with an example. [5 marks]

### Answer

A transmission error changes one or more bits. A **single-bit error** flips one bit. A **burst error** affects a span from the first corrupted bit to the last; the bits in between need not all change. Detection identifies an invalid received pattern; correction identifies the intended pattern within the code’s guaranteed capability.

## Block coding

1. Divide data into fixed k-bit datawords. Map each one to an n-bit codeword, with n greater than k.
2. The n−k redundant bits create separation between valid codewords. There are 2k valid codewords, a subset of the 2n possible n-bit patterns, for an injective binary block code.
3. A receiver can reject patterns that are not valid codewords. Nearest-codeword decoding can correct errors only if the received pattern has an unambiguous nearest intended codeword under the error bound.
4. Error-control protocols may combine detection with retransmission (ARQ), or use sufficient coding redundancy for forward error correction (FEC).



$$
R_c=k/n,\quad r=n-k
$$



#### Practical example

A simple even-parity code maps 101 to 1010. The redundant parity bit permits detection of any single-bit flip, but cannot identify which bit changed.

**Yaad rakho:** Detection asks error hua? Correction asks kaunsa correct word?

**Hamming distance** d(a,b) is the number of positions where two equal-length bit strings differ. XOR the strings and count the 1s. **Minimum distance** dmin is the smallest distance between distinct valid codewords; do not use the largest pairwise distance.



$$
d(a,b)=\mathrm{weight}(a\oplus b)
$$





$$
s_{\max}=d_{\min}-1,\qquad t_{\max}=\left\lfloor\frac{d_{\min}-1}{2}\right\rfloor
$$



1. Detection: fewer than dmin flips cannot transform one valid codeword into another, so all such errors are detected.
2. Correction: radius-t error regions around different codewords must not overlap. Therefore 2t < dmin, or dmin ≥ 2t+1.
3. These are separate maximum guaranteed capabilities. A code with dmin=3 detects up to two errors in detection mode and corrects one in correction mode; that statement does not promise simultaneous correction of one and reliable identification of every two-error event under the same nearest-codeword decoder.

#### Practical example

The repetition code {000,111} has distance 3. If 001 arrives and at most one flip occurred, 000 is the unique nearest codeword.

**Yaad rakho:** XOR ones count; minimum minus one detect, half floor correct.