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)

This polynomial G(x) is degree 3, giving a 3-bit parity field. (See factors of G(x))

The systematic 7-bit codewords will have 4 data bits and 3 parity bits.


Generator Matrix (G) for (7, 4) Code

G is a (4 × 7) matrix

For 4-bit input data i the corresponding 7-bit codeword is given by C = iG.

The steps involved in using G(x) to create a generator matrix (G) for a systematic code are shown below.

STEP ONE  -  Creating a non-systematic generating matrix G

A suitable (4 × 7) generator matrix can be constructed by writing the generator polynomial 1011 shifted right one bit for each successive row. There are 4 rows corresponding to the choice of 4 input data bits. The 4 rows are labeled (0 to 3) for reference. While this is a valid generator polynomial, it does not generate a systematic code.

0: 1011000
1: 0101100
2: 0010110
3: 0001011

Each row in this generator matrix is also a valid 7-bit codeword, being divisible by P(x).

STEP TWO  -  Creating a systematic generating matrix   G = [Ik|P]

A systematic generator matrix G is distinguished by having a (k × k) identity matrix, often on the lefthand side as G = [Ik|P] such that each codeword includes (k) data bits followed by (n-k) parity bits. Alternatively, the format G = [P|Ik] places the identity matrix on the righthand side.

To adjust the above generator matrix for a (7,4) systematic code, the leftmost columns will be manipulated to form a (4 × 4) identity matrix. To this end, begin with the top row and add to it selected rows until the first 4-bits describe the top row of an identity matrix. The new Row 0 is formed by the sum of rows (3,2,0) refering to the labels shown above. Go down the matrix for each row until the complete generator matrix is formed.

1000101 = ROW(3+2+0)
0100111 = ROW(3+1)
0010110 = ROW(2)
0001011 = ROW(3)

Each row in the generator matrix is a valid 7-bit codeword. In a linear code, the sum of codewords is another valid codeword.


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

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

For example, rotating the 7-bit codeword (01) left by one bit gives the codeword (02):

(01) = 0001011
(02) = 0010110

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

When codewords are linear, any linear combination of codewords is another codeword. For example, the 7-bit codeword (01) is the sum (02)+(03)

(02) = 0010110
(03) = 0011101
(01) = 0001011


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.

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.


Check Matrix (H) for (7, 4) code

H is a (3 × 7) matrix

The check matrix for a systematic code can be found directly from the generator matrix. The rightmost columns form a (3 × 3) identity matrix while the remaining columns can be identified in corresponding rows in the generator matrix.

1110100
0111010
1101001


Each row in the check matrix is a valid 7-bit codeword only for a cyclic code.



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)


2024-03-04 14:51:41 AST
Last Updated: 2015-02-06
Richard Tervo [ tervo@unb.ca ] Back to the course homepage...