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) = x10+x8+x5+x4+x2+x+1

(10100110111)

Sample (15,5) Codewords  (cyclic)

DATA = 00 : 000000000000000
DATA = 01 : 000010100110111
DATA = 02 : 000101001101110
DATA = 03 : 000111101011001
DATA = 04 : 001000111101011
DATA = 05 : 001010011011100
DATA = 06 : 001101110000101
DATA = 07 : 001111010110010
DATA = 08 : 010001111010110
DATA = 09 : 010011011100001
DATA = 10 : 010100110111000
DATA = 11 : 010110010001111
DATA = 12 : 011001000111101
DATA = 13 : 011011100001010
DATA = 14 : 011100001010011
DATA = 15 : 011110101100100
more →

Distance Analysis

This sample subset of 15-bit codewords has a minimum distance D=7, correcting up to t=3 errors.

 00010203040506070809101112131415
00--070708080707080807070808070708
0107--0807070808070708080707080807
020708--07070808070708080707080807
03080707--080707080807070808070708
0408070708--0707080807070808070708
050708080707--08070708080707080807
06070808070708--070708080707080807
0708070708080707--0807070808070708
080807070808070708--07070808070708
09070808070708080707--080707080807
1007080807070808070708--0707080807
110807070808070708080707--08070708
12080707080807070808070708--070708
1307080807070808070708080707--0807
140708080707080807070808070708--07
15080707080807070808070708080707--

This sampling of 16 codewords is not necessarily indicative of the error control performance of all 25 = 32 possible codewords.


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)

2024-04-29 19:15:45 ADT
Last Updated: 2015-02-06 		Richard Tervo [ tervo@unb.ca ] Back to the course homepage...