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) = x4+x+1

(10011)

This polynomial G(x) is degree 4, giving a 4-bit parity field. (See factors of G(x))

The systematic 15-bit codewords will have 11 data bits and 4 parity bits.

Sample (15,11) Codewords  (cyclic)

When codewords are cyclic the circular shift of a valid codeword produces another valid codeword.

For example, rotating the 15-bit codeword (01) left by one bit gives the codeword (02):

(01) = 000000000010011
(02) = 000000000100110

DATA = 00 : 000000000000000
DATA = 01 : 000000000010011
DATA = 02 : 000000000100110
DATA = 03 : 000000000110101
DATA = 04 : 000000001001100
DATA = 05 : 000000001011111
DATA = 06 : 000000001101010
DATA = 07 : 000000001111001
DATA = 08 : 000000010001011
DATA = 09 : 000000010011000
DATA = 10 : 000000010101101
DATA = 11 : 000000010111110
DATA = 12 : 000000011000111
DATA = 13 : 000000011010100
DATA = 14 : 000000011100001
DATA = 15 : 000000011110010
more →

When codewords are linear, any linear combination of codewords is another codeword. For example, the 15-bit codeword (01) is the sum (02)+(03)

(02) = 000000000100110
(03) = 000000000110101
(01) = 000000000010011

Distance Analysis

This sample subset of 15-bit codewords has a minimum distance D=3, correcting up to t=1 error.

 00010203040506070809101112131415
00--030304030604050403050605040405
0103--0403060305040304060504050504
020304--03040503060506040304050504
03040303--050406030605030405040405
0403060405--0303040504040504030506
050603050403--04030405050403040605
06040503060304--030405050405060403
0705040603040303--0504040506050304
080403050605040405--03030403060405
09030406050405050403--040306030504
1005060403040505040304--0304050306
110605030405040405040303--05040603
12050404050403050603060405--030304
1304050504030406050603050403--0403
140405050405060403040503060304--03
15050404050605030405040603040303--

This sampling of 16 codewords is not necessarily indicative of the error control performance of all 211 = 2048 possible codewords.

To determine the minimum distance between any two codewords in a linear block code, it is sufficient to check every codeword once against the all-zero codeword. In other words, the Hamming distance of a code may be determined from the distances in row 00 only. Moreover, the distance from a given codeword to zero is found by the sum of the 1's in the codeword. The remaining codewords could be easily checked in this way.


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 15:58:43 ADT
Last Updated: 2015-02-06 		Richard Tervo [ tervo@unb.ca ] Back to the course homepage...