UNB ECE4253 Digital Communications
Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

Polynomial Code Generator Tool

Given a generator polynomial G(x) of degree p and a binary input data size k, this online tool creates and displays a generator matrix G, a check matrix H, and a demonstration of the resulting systematic codewords for this (n,k) code, where n=p+k. The nature of G(x) and the value of k will determine the utility of the codewords in a error control scheme.

The mathematics of error control can be based on either a matrix or a polynomial approach. This page shows how any polynomial G(x) may be used to define an equivalent check matrix and generator matrix. Conversely, it is not always possible to find a polynomial G(x) corresponding to an arbitary generator matrix.

The generator polynomial G(x) can be up to degree p=36, and the input data size is limited to k=36 bits.

Polynomial G(x)

G(x) = x3+x+1

(1011)

Complete Set of (7,4) Codewords  (cyclic)

DATA = 00 : 0000000
DATA = 01 : 0001011
DATA = 02 : 0010110
DATA = 03 : 0011101
DATA = 04 : 0100111
DATA = 05 : 0101100
DATA = 06 : 0110001
DATA = 07 : 0111010
DATA = 08 : 1000101
DATA = 09 : 1001110
DATA = 10 : 1010011
DATA = 11 : 1011000
DATA = 12 : 1100010
DATA = 13 : 1101001
DATA = 14 : 1110100
DATA = 15 : 1111111

Distance Analysis

This complete set of 7-bit codewords has a minimum distance D=3, correcting up to t=1 error.

 00010203040506070809101112131415
00--030304040303040304040303040407
0103--0403030404030403030404030704
020304--03030404030403030404070304
03040303--040303040304040307040403
0404030304--0303040304040703040403
050304040303--04030403070404030304
06030404030304--030407030404030304
0704030304040303--0704040303040403
080304040303040407--03030404030304
09040303040403070403--040303040403
1004030304040703040304--0303040403
110304040307040403040303--04030304
12030404070304040304030304--030304
1304030704040303040304040303--0403
140407030404030304030404030304--03
15070404030304040304030304040303--

All 16 available codewords have been verifed in this table.


Specify a new polynomial or a different number of data bits.

Model M20J GENERATOR POLYNOMIAL TOOL
Data Bits k =   G(x):

Discussion Codewords Generator Format
G = [Ik|P]
G = [P|Ik]
MATLAB Matrices

Examples

  1. (8,7) Simple Parity Bit (D=2) no error correction

  2. (7,4) Hamming Code (D=3) single bit error correction

  3. (15,11) Hamming Code (D=3) single bit error correction

  4. (15,10) Extended Hamming Code (D=4) single bit error correction

  5. (31,21) BCH Code (D=5) double bit error correction (notes)

  6. (15,5) BCH Code (D=7) triple bit error correction (notes)

  7. (23,12,7) Binary Golay Code (D=7) triple bit error correction

  8. (35,27) Fire Code specialized 3-bit burst error correction

  9. 16-bit CRC (CCITT) commonly used for error detection (notes)

2024-04-25 18:52:19 ADT
Last Updated: 2015-02-06 		Richard Tervo [ tervo@unb.ca ] Back to the course homepage...