--- title: "(7,4) cyclic code g(x) = 1 + x + x³ for 1011" subject: "Communication Systems" chapter: 8 key: "semester_6/communication_systems:cyclic-7-4-1x-x3-1011" source: https://www.bctengineer.com/q/semester_6/communication_systems/cyclic-7-4-1x-x3-1011 site: BCT Engineering --- # (7,4) cyclic code g(x) = 1 + x + x³ for 1011 Communication Systems · Chapter 8: Error Detection and Correction Coding - Asked in: 2070 Chaitra (CS II), 6 marks - Video: [Problem on systematic and non-systematic code vectors](https://www.youtube.com/watch?v=q3OUAXpGPCo) — PRAKASAM TUTORIALS - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/cyclic-7-4-1x-x3-1011 - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-8 ## Question The generator polynomial of a (7,4) cyclic code is g(x) = 1 + x + x³. Find the code for the message vector 1011 in a non-systematic and systematic form. ## Short answer Non-systematic: multiply m(x)=1+x²+x³ by g(x)=1+x+x³ modulo 2, giving codeword 1111111. Systematic: divide x³m(x) by g(x); remainder 1 yields parity bits 100, codeword 1001011. ## Answer A **cyclic code** is a linear block code in which every cyclic shift of a codeword is also a codeword. An $(n,k)$ cyclic code is generated by a polynomial $g(x)$ of degree $n-k$ that divides $x^n+1$: - **Non-systematic encoding:** $c(x)=m(x)\,g(x)$; message bits are not visible directly in the codeword. - **Systematic encoding:** $c(x)=x^{n-k}m(x)+\text{rem}\left[\dfrac{x^{n-k}m(x)}{g(x)}\right]$; the message appears unchanged and the remainder gives the $n-k$ parity bits. Given $(n,k)=(7,4)$, $g(x)=1+x+x^3$, $n-k=3$. Convention (Haykin): message $(m_0m_1m_2m_3)=1011$ in ascending powers of $x$: $$ m(x)=1+x^2+x^3 $$ All arithmetic is modulo 2. #### Non-systematic form $$ \begin{aligned} c(x)&=m(x)g(x)=(1+x^2+x^3)(1+x+x^3)\\ &=(1+x+x^3)+(x^2+x^3+x^5)+(x^3+x^4+x^6)\\ &=1+x+x^2+x^3+x^4+x^5+x^6 \end{aligned} $$ ($x^3$ appears three times, leaving $x^3$.) **Non-systematic code vector $(c_0\dots c_6)$ = 1111111** #### Systematic form 1. $x^{n-k}m(x)=x^3+x^5+x^6$ 2. Divide by $g(x)=x^3+x+1$: ```text x^3 + x^2 + x + 1 --------------------------- x^3+x+1 ) x^6 + x^5 + x^3 x^6 + x^4 + x^3 ------------------------- x^5 + x^4 x^5 + x^3 + x^2 ----------------------- x^4 + x^3 + x^2 x^4 + x^2 + x ------------------- x^3 + x x^3 + x + 1 ----------------- 1 ``` 3. Remainder $b(x)=1$, parity bits $(b_0b_1b_2)=100$. 4. $c(x)=b(x)+x^3m(x)=1+x^3+x^5+x^6$. Check: quotient $\times\,g(x)=(1+x+x^2+x^3)(1+x+x^3)=1+x^3+x^5+x^6$. Valid. **Systematic code vector $(b_0b_1b_2\,m_0m_1m_2m_3)$ = 100 1011** | Form | Code vector | |---|---| | Non-systematic | 1111111 | | Systematic | 1001011 | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the Haykin convention, what is m(x) for message 1011? A. x^3 + x + 1 B. 1 + x + x^3 C. 1 + x^2 + x^3 D. x^2 + x + 1 Answer: C. Bits m0..m3 = 1,0,1,1 in ascending powers give 1 + x^2 + x^3. (Key point: Polynomial convention) 2. What is c(x) = m(x)g(x) and the vector? A. 1 + x^3 + x^5 + x^6, giving 1001011 B. 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, giving 1111111 C. x^2 + x^3 + x^5, giving 0011010 D. 1 + x^2, giving 1010000 Answer: B. (1+x^2+x^3)(1+x+x^3) sums to all seven powers (x^3 appears three times leaving one) mod 2. (Key point: Non-systematic result) 3. What is the remainder of x^3 m(x) divided by g(x), and the parity bits? A. Remainder 1, parity b0b1b2 = 100 B. Remainder x^2, parity 001 C. Remainder x, parity 010 D. Remainder 0, parity 000 Answer: A. x^3+x^5+x^6 divided by x^3+x+1 gives quotient 1+x+x^2+x^3 and remainder 1. (Key point: Systematic parity) 4. What is the systematic code vector? A. 1011100 B. 1111111 C. 0011101 D. 1001011 (parity 100 then message 1011) Answer: D. c(x) = b(x) + x^3 m(x) = 1 + x^3 + x^5 + x^6, arranged as parity bits then message. (Key point: Systematic vector layout) ## Close variants set in other exams - [(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× ## 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× - [(7,4) cyclic code with g(x) = x³ + x² + 1 for 1011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acyclic-code-7-4-1011-x3x2) — Communication Systems, chapter 8, asked 2× - [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 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× - [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 - [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 - [Cyclic Redundancy Code with example](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Acrc-example) — Instrumentation II, chapter 3 - [Mass and energy equations for non-flow; cyclic process](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Amass-energy-nonflow-cyclic-closed) — Fundamental of Thermodynamics & Heat Transfer, chapter 4