--- title: "Hamming codes and their properties" subject: "Data Communication" chapter: 8 key: "semester_5/data_communication:hamming-code-properties" source: https://www.bctengineer.com/q/semester_5/data_communication/hamming-code-properties site: BCT Engineering --- # Hamming codes and their properties Data Communication · Chapter 8: Information Theory and Coding - Asked in: 2072 Chaitra, 2+4+4 marks - Video: [Hamming Code | Error Detection](https://www.youtube.com/watch?v=1A_NcXxdoCc) — Neso Academy - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/data_communication/hamming-code-properties - Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-8 ## Question What are Hamming codes? Write the properties of Hamming codes. Visualize a 3-bit code words as code vector. ## Short answer Hamming codes are perfect linear block codes with d_min=3, correcting single-bit errors. For r parity bits, n=2^r-1, k=n-r. 3-bit words form a cube; repetition code {000,111} has d_min=3, even-parity code detects 1 error. ## Answer #### Hamming codes **Hamming codes** are linear block codes that can detect up to two-bit errors and correct all single-bit errors. For r ≥ 2 parity bits the code is (n, k) = (2ʳ − 1, 2ʳ − 1 − r), e.g. (7,4) and (15,11). The parity bits are placed at positions 1, 2, 4, 8, … and each checks the positions that contain it in their binary index. #### Properties - Block length n = 2ʳ − 1 and k = n − r data bits; rate k/n = 1 − r/n approaches 1 for large r. - Minimum Hamming distance d_min = 3: detects 2 errors or corrects 1 error. - They are **perfect codes**: all 2ⁿ words lie within distance 1 of exactly one codeword. - Linear (sum of codewords is a codeword), and can be made cyclic. - Syndrome decoding: the r-bit syndrome gives the exact position of a single-bit error (0 = no error). - Adding an overall parity bit gives an extended Hamming code with d_min = 4 (single-error correction, double-error detection). - Simple to implement with XOR gates; used in ECC memory. #### Visualising 3-bit words as code vectors All 3-bit words are the 8 vertices of a cube; the Hamming distance between two words is the number of edges on the shortest path between them. ```text 011 ---------- 111 /| /| / | / | 010 ---------- 110 | | 001 -------|-- 101 | / | / |/ |/ 000 ---------- 100 ``` - Repetition code {000, 111}: the two vertices are 3 edges apart, d_min = 3, corrects 1 error (any word one step from 000 decodes to 000, one step from 111 decodes to 111). - Even-parity code {000, 011, 101, 110}: every pair is 2 edges apart, d_min = 2, detects 1 error but cannot correct it. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What can a Hamming code do? A. Detect up to two-bit errors and correct all single-bit errors B. Correct all two-bit errors C. Only detect single-bit errors D. Correct burst errors of any length Answer: A. Hamming codes have d_min = 3, so they detect up to two-bit errors and correct single-bit errors. (Key point: Hamming code capability) 2. For r parity bits, what are n and k of a Hamming code? A. n = 2r, k = r B. n = 2^r - 1, k = 2^r - 1 - r C. n = r^2, k = r D. n = 2^r, k = 2^r - r Answer: B. A Hamming code has n = 2^r - 1 and k = n - r, e.g. (7,4) for r = 3. (Key point: Block length) 3. What does the syndrome of a received Hamming word indicate? A. Number of errors in the word B. Position of a single-bit error (0 means no error) C. The parity of the data bits only D. The code rate Answer: B. The r-bit syndrome read as a binary number gives the position of the erroneous bit, 0 meaning no error. (Key point: Syndrome decoding) 4. On the cube of 3-bit words, what is the repetition code {000, 111}'s d_min and what does that mean? A. d_min = 1, no protection B. d_min = 2, detects 1 error only C. d_min = 3, corrects 1 error D. d_min = 3, corrects 2 errors Answer: C. 000 and 111 are opposite corners, 3 edges apart, so d_min = 3 and any single-error word decodes to the nearer codeword. (Key point: Cube representation) ## Related in Data Communication - [(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 - [Short note: Hamming codes](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahamming-codes-short-note) — Data Communication, chapter 8 - [All code words of the (6,3) code with given G](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ablock-code-63-all-codewords) — Data Communication, chapter 8 - [Construct a (7,4) Hamming code](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahamming-74-construct) — Data Communication, chapter 8 - [Forward error correction using Hamming code](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afec-hamming) — Data Communication, chapter 8 - [Trellis decoding of 1/2 rate conv. code (3-bit errors)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconv-trellis-3bit) — Data Communication, chapter 8, asked 2× ## Related in other subjects - [Speech characteristics and 7-bit Hamming code](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aspeech-hamming-code) — Wireless Communication, chapter 6 - [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× - [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 - [Hamming weight and distance; validate code 100011](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahamming-6-3-h-100011) — Communication Systems, chapter 8