---
title: "Convolutional vs block coder; (7,4) cyclic code for 1010"
subject: "Communication Systems"
chapter: 8
key: "semester_6/communication_systems:conv-vs-block-cyclic-7-4-1010"
source: https://www.bctengineer.com/q/semester_6/communication_systems/conv-vs-block-cyclic-7-4-1010
site: BCT Engineering
---

# Convolutional vs block coder; (7,4) cyclic code for 1010

Communication Systems · Chapter 8: Error Detection and Correction Coding

- Asked in: 2073 Chaitra (CS II), 2+6 marks
- Video: [SYSTEMATIC CYCLIC CODE with SOLVED EXAMPLE -Digital Communication | Electronics Subjectified | Hindi](https://www.youtube.com/watch?v=mXUXBsfpcMA) — Electronics Subjectified
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/conv-vs-block-cyclic-7-4-1010
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-8

## Question

Why convolution coder are better suited than block coder? Determine systematic and non-systematic code vector for a (7,4) cyclic hamming code for message vector {1010} with generator polynomial g(x) = 1 + x + x³

## Short answer

Convolutional coders handle continuous streams with memory and enable efficient soft-decition Viterbi decoding, unlike block coders. For (7,4) code with g(x)=1+x+x^3 and message 1010, non-systematic: 1110010; systematic: 0011010.

## Answer

#### Why convolutional coders are better suited than block coders

- **Continuous operation:** they encode a serial bit stream directly; block coders must wait for $k$ bits, adding delay and buffering.
- **Memory:** each output depends on the present and $K-1$ previous bits, so redundancy is spread over a long span, giving good performance against random noise.
- **Soft-decision decoding:** the Viterbi algorithm easily uses soft (analog) values, giving about 2 dB extra gain, which is hard for algebraic block decoders.
- **Simple hardware:** shift registers and XOR gates; widely used in satellite, GSM and Wi-Fi links.

#### (7,4) cyclic Hamming code, $g(x)=1+x+x^3$, message 1010

Convention (Haykin): $(m_0m_1m_2m_3)=1010$, ascending powers:

$$
m(x)=1+x^2
$$

**Non-systematic:**

$$
\begin{aligned}
c(x)&=m(x)g(x)=(1+x^2)(1+x+x^3)\\
&=1+x+x^3+x^2+x^3+x^5\\
&=1+x+x^2+x^5
\end{aligned}
$$

$(c_0c_1\dots c_6)$ = **1110010**

**Systematic:**

1. $x^{n-k}m(x)=x^3(1+x^2)=x^3+x^5$
2. Divide by $g(x)$:

```text
              x^2
          ------------------
x^3+x+1 ) x^5 +       x^3
          x^5 +       x^3 + x^2
          ---------------------
                            x^2   <- remainder
```

3. $b(x)=x^2$, parity bits $(b_0b_1b_2)=001$.
4. $c(x)=b(x)+x^3m(x)=x^2+x^3+x^5$.

$(b_0b_1b_2\,m_0m_1m_2m_3)$ = **001 1010**

Check: $c(x)=x^2g(x)=x^2+x^3+x^5$, divisible by $g(x)$, so valid.

| Form | Code vector |
|---|---|
| Non-systematic | 1110010 |
| Systematic | 0011010 |

## Check yourself

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

1. Which is a reason convolutional coders are better suited than block coders?
   A. They need no memory
   B. They encode a continuous serial stream without waiting for k bits
   C. They cannot use soft decisions
   D. They always have higher rate
   Answer: B. Convolutional coders work on the stream directly, avoiding block buffering delay, and their memory spreads redundancy. (Key point: Advantage of convolutional codes)

2. What extra benefit does Viterbi decoding of convolutional codes give?
   A. Easy use of soft decisions with about 2 dB gain
   B. Zero decoding delay
   C. No need for a trellis
   D. Removal of the need for a generator
   Answer: A. The Viterbi algorithm readily uses soft values, giving about 2 dB extra gain. (Key point: Soft-decision decoding)

3. With ascending-power convention, m(x) = 1 + x^2 and g(x) = 1 + x + x^3; what is the non-systematic code vector?
   A. 0011010
   B. 1011100
   C. 1001011
   D. 1110010
   Answer: D. (1+x^2)(1+x+x^3) = 1 + x + x^2 + x^5, giving c0..c6 = 1110010. (Key point: Non-systematic cyclic code)

4. In this convention, what is the systematic vector and its structure?
   A. 1010 001: message first, then parity
   B. 1110010
   C. 001 1010: parity bits b0b1b2 first, then message
   D. 0101 100
   Answer: C. x^3 m(x) = x^3 + x^5 leaves remainder x^2, so b = 001, placed before the message 1010. (Key point: Systematic cyclic code)

## Close variants set in other exams

- [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×

## Related in Communication Systems

- [Hamming weight and distance; (7,4) cyclic code for 1011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahamming-weight-cyclic-1011) — Communication Systems, chapter 8, asked 2×
- [Significance of dmin; (7,4) cyclic code for 1101 with g(x) = 1 + x + x²](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Admin-cyclic-7-4-1101) — Communication Systems, chapter 8
- [(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×
- [(7,4) cyclic code g(x) = 1 + x + x³ for 1011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acyclic-7-4-1x-x3-1011) — Communication Systems, chapter 8, asked 2×
- [Short note: Syndrome calculation in linear systematic block code](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-syndrome-calculation) — Communication Systems, chapter 8

## Related in other subjects

- [Redundancy, Hamming distance, linear vs cyclic](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aredundancy-hamming-linear-cyclic) — Data Communication, chapter 8