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) = x11+x10+x6+x5+x4+x2+1

(110001110101)

This polynomial G(x) is degree 11, giving a 11-bit parity field. (See factors of G(x))

The systematic 23-bit codewords will have 12 data bits and 11 parity bits.

Sample (23,12) Codewords  (cyclic)

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

For example, rotating the 23-bit codeword (01) left by one bit gives the codeword (02):

(01) = 00000000000110001110101
(02) = 00000000001100011101010

DATA = 00 : 00000000000000000000000
DATA = 01 : 00000000000110001110101
DATA = 02 : 00000000001010010011111
DATA = 03 : 00000000001100011101010
DATA = 04 : 00000000010010101001011
DATA = 05 : 00000000010100100111110
DATA = 06 : 00000000011000111010100
DATA = 07 : 00000000011110110100001
DATA = 08 : 00000000100011011100011
DATA = 09 : 00000000100101010010110
DATA = 10 : 00000000101001001111100
DATA = 11 : 00000000101111000001001
DATA = 12 : 00000000110001110101000
DATA = 13 : 00000000110111111011101
DATA = 14 : 00000000111011100110111
DATA = 15 : 00000000111101101000010
DATA = 16 : 00000001000000110110011
DATA = 17 : 00000001000110111000110
DATA = 18 : 00000001001010100101100
DATA = 19 : 00000001001100101011001
DATA = 20 : 00000001010010011111000
DATA = 21 : 00000001010100010001101
DATA = 22 : 00000001011000001100111
DATA = 23 : 00000001011110000010010
DATA = 24 : 00000001100011101010000
DATA = 25 : 00000001100101100100101
DATA = 26 : 00000001101001111001111
DATA = 27 : 00000001101111110111010
DATA = 28 : 00000001110001000011011
DATA = 29 : 00000001110111001101110
DATA = 30 : 00000001111011010000100
DATA = 31 : 00000001111101011110001
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When codewords are linear, any linear combination of codewords is another codeword. For example, the 23-bit codeword (01) is the sum (02)+(03)

(02) = 00000000001010010011111
(03) = 00000000001100011101010
(01) = 00000000000110001110101

Distance Analysis

This sample subset of 23-bit codewords has a minimum distance D=7, correcting up to t=3 errors.

 0001020304050607080910111213141516171819202122232425262728293031
00--07080707080708080708070712110807080708080708070708111208110811
0107--070808070807070807081207081108070807070807080807121111081108
020807--0707080708080708071108071207080708080708071112070808110811
03070807--08070807070807080811120708070807070807081211080711081108
0407080708--070807071211080807080708070807070807080811081107081112
050807080707--0708120708110708070807080708080708071108110808071211
06070807080807--07110807120807080708070807070807080811081111120708
0708070807070807--081112070708070807080708080708071108110812110807
080807080707121108--0708070708070807081112081108110708070808070807
09070807081207081107--07080807080708071211110811080807080707080708
1008070807110807120807--070708070811120708081108110708070808070807
110708070808111207070807--0807080712110807110811080807080707080708
12071211080807080707080708--07080708110811070811120807080707080708
1312070811070807080807080707--070811081108080712110708070808070807
141108071208070807070807080807--0708110811111207080807080707080708
15081112070708070808070807070807--11081108121108070708070808070807
1607080708080708070708111208110811--070807070807080807080707121108
170807080707080708080712111108110807--0708080708070708070812070811
18070807080807080711120708081108110807--07070807080807080711080712
1908070807070807081211080711081108070807--080708070708070808111207
200807080707080708081108110708111207080708--0708070712110808070807
21070807080807080711081108080712110807080707--07081207081107080708
2208070807070807080811081111120708070807080807--071108071208070807
230708070808070807110811081211080708070807070807--0811120707080708
24070811120811081107080708080708070807080707121108--07080707080708
2508071211110811080807080707080708070807081207081107--070808070807
261112070808110811070807080807080708070807110807120807--0707080708
27121108071108110808070807070807080708070808111207070807--08070807
2808110811070811120807080707080708071211080807080707080708--070807
291108110808071211070807080807080712070811070807080807080707--0708
30081108111112070808070807070807081108071208070807070807080807--07
3111081108121108070708070808070807081112070708070808070807070807--

This sampling of 32 codewords is not necessarily indicative of the error control performance of all 212 = 4096 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)

2024-05-16 03:43:49 ADT
Last Updated: 2015-02-06 		Richard Tervo [ tervo@unb.ca ] Back to the course homepage...