---
title: "Short note: Syndrome calculation in linear systematic block code"
subject: "Communication Systems"
chapter: 8
key: "semester_6/communication_systems:short-note-syndrome-calculation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-syndrome-calculation
site: BCT Engineering
---

# Short note: Syndrome calculation in linear systematic block code

Communication Systems · Chapter 8: Error Detection and Correction Coding

- Asked in: 2069 Chaitra (CS II), 4 marks
- Video: [Error Syndromes & Error Correction in Linear Block Code in Digital Communication](https://www.youtube.com/watch?v=T7mSu62-RLE) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-syndrome-calculation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-8

## Question

Write a short note on syndrome calculation in linear systematic block code.

## Short answer

Syndrome s = r H^T equals the XOR of parity bits recomputed from the received message and the received parity bits. s=0 means no detectable error; s≠0 indicates an error, and for a single error s matches the corresponding column of H.

## Answer

In a linear systematic $(n,k)$ block code, $G=[I_k\mid P]$ and the parity-check matrix is $H=[P^T\mid I_{n-k}]$, so every codeword obeys $\mathbf{c}H^T=\mathbf{0}$. The **syndrome** of a received vector $\mathbf{r}$ is the $(n-k)$-bit vector

$$
\mathbf{s}=\mathbf{r}H^T
$$

#### Properties

- Writing $\mathbf{r}=\mathbf{c}\oplus\mathbf{e}$: $\mathbf{s}=\mathbf{e}H^T$, i.e. it depends only on the error pattern.
- $\mathbf{s}=\mathbf{0}$: no detectable error. $\mathbf{s}\neq\mathbf{0}$: error present.
- For a single error in bit $i$, $\mathbf{s}$ = $i$-th column of $H$, so the error is located directly.
- All error patterns in one coset of the standard array share one syndrome; the coset leader is taken as the error.

#### Calculation

In systematic form, $\mathbf{s}$ = (parity bits recomputed from the received message bits) $\oplus$ (received parity bits).

```text
 r = [ received msg | received parity ]
         |                 |
   [recompute parity]      |
         |                 |
         +------(XOR)------+--> syndrome s
```

Example, $(6,3)$ code with $H=\begin{bmatrix}1&0&1&1&0&0\\0&1&1&0&1&0\\1&1&0&0&0&1\end{bmatrix}$ and $\mathbf{r}=100011$:

$$
\mathbf{s}=\text{col}_1\oplus\text{col}_5\oplus\text{col}_6=(1,1,0)
$$

This equals column 3, so bit 3 is wrong; corrected word 101011. For cyclic codes, the syndrome is the remainder of $r(x)/g(x)$, computed with a shift register.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the syndrome of a received vector r defined for a systematic code?
   A. s = rG, a k-bit vector
   B. s = r + G
   C. s = rH^T, an (n-k)-bit vector
   D. s = number of 1s in r
   Answer: C. The syndrome is an (n-k)-bit vector s = rH^T. (Key point: Syndrome definition)

2. What does a nonzero syndrome mean and what does a single-bit error give?
   A. No error; s equals the message
   B. An error is present; s equals the i-th column of H for an error in bit i
   C. Two errors; s equals the codeword
   D. An error in parity only
   Answer: B. s = eH^T, so a single error at bit i gives column i of H. (Key point: Meaning of s)

3. In systematic form, how can s be computed?
   A. Recompute parity from received message bits and XOR with received parity bits
   B. Add the received message bits only
   C. Multiply r by the message
   D. Count the weight of r
   Answer: A. For H = [P^T | I], s = (message x P) XOR received parity. (Key point: Systematic shortcut)

4. For a cyclic code, how is the syndrome obtained?
   A. As the quotient of r(x) by g(x)
   B. As r(x) times g(x)
   C. As the first k bits of r
   D. As the remainder of r(x) divided by g(x)
   Answer: D. The cyclic-code syndrome is rem[r(x)/g(x)], computed with a shift register. (Key point: Cyclic codes)

## Related in Communication Systems

- [Convolutional vs block codes; (7,4) cyclic code for 1011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aconv-vs-block-cyclic-7-4-1011) — Communication Systems, chapter 8, asked 2×
- [Convolutional vs block coder; (7,4) cyclic code for 1010](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aconv-vs-block-cyclic-7-4-1010) — Communication Systems, chapter 8, asked 2×
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- [Hamming distance and weight importance; syndrome decoding](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahamming-importance-syndrome-decoding) — Communication Systems, chapter 8
- [Analog and digital system block diagrams; linear and non-linear distortion](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-digital-block-linear-nonlinear-distortion) — Communication Systems, chapter 1
- [(7,4) cyclic code G(p) = p³ + p + 1 for 0101](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acyclic-7-4-p3p1-0101) — Communication Systems, chapter 8, asked 2×

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