---
title: "Significance of dmin; (7,4) cyclic code for 1101 with g(x) = 1 + x + x²"
subject: "Communication Systems"
chapter: 8
key: "semester_6/communication_systems:dmin-cyclic-7-4-1101"
source: https://www.bctengineer.com/q/semester_6/communication_systems/dmin-cyclic-7-4-1101
site: BCT Engineering
---

# Significance of dmin; (7,4) cyclic code for 1101 with g(x) = 1 + x + x²

Communication Systems · Chapter 8: Error Detection and Correction Coding

- Asked in: 2081 Bhadra (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/dmin-cyclic-7-4-1101
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-8

## Question

What is the significance of (dmin) minimum hamming distance? Determine the systematic and non-systematic code vector for a (7,4) cyclic code for message vector (m₀, m₁, m₂, m₃) = (1101) for a given generator polynomial g(x) = 1 + x + x².

## Short answer

Minimum Hamming distance d_min determines error detection (up to d_min-1) and correction (up to floor((d_min-1)/2)) capability. For (7,4) cyclic code with message 1101 and generator 1+x+x^3 (corrected), non-systematic vector is 1010001, systematic is 0001101.

## Answer

#### Significance of $d_{min}$

The **minimum Hamming distance** $d_{min}$ is the smallest number of positions in which any two codewords differ (for a linear code, the smallest non-zero codeword weight). It fixes the power of the code:

- Detects up to $s$ errors if $d_{min}\ge s+1$.
- Corrects up to $t$ errors if $d_{min}\ge 2t+1$, i.e. $t=\lfloor(d_{min}-1)/2\rfloor$.

For a (7,4) Hamming code $d_{min}=3$: detects 2, corrects 1 error.

#### Note on the generator polynomial

A generator for a (7,4) cyclic code must have degree $n-k=3$ and divide $x^7+1=(1+x)(1+x+x^3)(1+x^2+x^3)$. The printed $g(x)=1+x+x^2$ has degree 2 and does not divide $x^7+1$, so it is taken as a misprint of the standard **$g(x)=1+x+x^3$**.

Message $(m_0m_1m_2m_3)=1101$, ascending powers:

$$
m(x)=1+x+x^3
$$

#### Non-systematic code vector

$$
\begin{aligned}
c(x)&=m(x)g(x)=(1+x+x^3)(1+x+x^3)\\
&=1+x^2+x^6\quad(\text{cross terms cancel in mod 2: }(a+b+c)^2=a^2+b^2+c^2)
\end{aligned}
$$

$(c_0c_1\dots c_6)$ = **1010001**

#### Systematic code vector

$$
x^3m(x)=x^3+x^4+x^6
$$

Divide by $g(x)=1+x+x^3$:

```text
              x^3
          -----------------------
x^3+x+1 ) x^6 + x^4 + x^3
          x^6 + x^4 + x^3
          ---------------
                          0
```

Remainder $b(x)=0$, parity bits $(b_0b_1b_2)=000$.

$$
c(x)=b(x)+x^3m(x)=x^3+x^4+x^6
$$

$(b_0b_1b_2\,m_0m_1m_2m_3)$ = **000 1101**

This happens because $m(x)$ itself equals $g(x)$, so $x^3m(x)$ is already a multiple of $g(x)$.

| Form | Code vector | Weight |
|---|---|---|
| Non-systematic | 1010001 | 3 |
| Systematic | 0001101 | 3 |

Both have weight 3, equal to $d_{min}$ of the code.

## Check yourself

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

1. If dmin = 3, how many errors can be corrected and how many detected?
   A. Correct 2, detect 3
   B. Correct 1, detect 2
   C. Correct 1, detect 3
   D. Correct 3, detect 1
   Answer: B. t = floor((3-1)/2) = 1 corrected and dmin-1 = 2 detected. (Key point: Significance of dmin)

2. Why is g(x) = 1 + x + x^2 not a valid generator for a (7,4) cyclic code?
   A. It has degree 2 instead of n-k = 3 and does not divide x^7 + 1
   B. It has too many terms
   C. It has odd weight
   D. It equals x^7 + 1
   Answer: A. A (7,4) generator must have degree 3 and divide x^7 + 1; the answer treats it as a misprint of 1 + x + x^3. (Key point: Valid generator)

3. For m(x) = 1 + x + x^3 and g(x) = 1 + x + x^3, what is c(x) = m(x)g(x) mod 2?
   A. 1 + x^2 + x^6, vector 1010001
   B. 1 + x + x^2 + x^3 + x^4 + x^5 + x^6
   C. x^3 + x^4 + x^6
   D. 1 + x^6, vector 1000001
   Answer: A. In mod 2, (a+b+c)^2 = a^2+b^2+c^2, so the square is 1 + x^2 + x^6. (Key point: Non-systematic code)

4. Why is the parity of the systematic code vector 000 for message 1101?
   A. Because the message has odd weight
   B. Because m(x) equals g(x), so x^3 m(x) is already divisible by g(x) (remainder 0)
   C. Because the quotient is zero
   D. Because the code is not cyclic
   Answer: B. x^3 m(x) = x^3 g(x) gives remainder 0, so the codeword is 000 1101. (Key point: Systematic code)

## 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×
- [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×
- [(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×
- [Hamming distance and weight; (6,3) code check for 100101](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahamming-6-3-h-100101) — Communication Systems, chapter 8

## Related in other subjects

- [Minimum Hamming distance; find k and n](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Amin-hamming-k-n) — Data Communication, chapter 8
- [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
- [(6,3) block code words, weights, dmin](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ablock-code-63-matrix-weights) — Data Communication, chapter 8
- [Capability of a code with dmin = 4](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahamming-distance-dmin4) — Data Communication, chapter 8