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

On this page, various polynomials can be chosen to demonstrate the structure of BCH codes capable of correcting several bit errors.

Creating a BCH Generator Polynomial

A BCH generating polynomial can produce codewords with predictable distance properties given a set of distinct minimal polynomials.

1. Let t be the number of bit errors to be corrected. The required Hamming distance is D=2t+1
2. Identify a set of minimal polynomials from {aⁿ} for n= 1 to 2t.
3. Reduce the set by deleting any duplicate minimal polynomials.
4. The BCH generating polynomial G(x) is the product of the remaining minimal polynomials.

This G(x) is the product of two minimal polynomials for {a¹,a³} as:

The original polynomial P(x) of degree 5 defines a codeword size of n=31 bits where 31 = 2⁵-1; the BCH polynomial G(x) of degree 10 defines 10 parity bits such that k = 31 - 10 = 21 to give (n,k) = (31,21).

This BCH (31,21) code can correct at least 2 bit errors, as specified. (See the codewords and matrices)

Model C-182T BCH CODE GENERATOR P(x) = 1011 : P[x] = x3+x+1 1101 : P[x] = x3+x2+1 10011 : P[x] = x4+x+1 11001 : P[x] = x4+x3+1 100101 : P[x] = x5+x2+1 101001 : P[x] = x5+x3+1 101111 : P[x] = x5+x3+x2+x+1 110111 : P[x] = x5+x4+x2+x+1 111011 : P[x] = x5+x4+x3+x+1 111101 : P[x] = x5+x4+x3+x2+1 1000011 : P[x] = x6+x+1 1011011 : P[x] = x6+x4+x3+x+1 1100001 : P[x] = x6+x5+1 1100111 : P[x] = x6+x5+x2+x+1 1101101 : P[x] = x6+x5+x3+x2+1 1110011 : P[x] = x6+x5+x4+x+1    t= 1 2 3 4 5 6 7 8 9 Derivation  | Row Table  | Discussion  | MATLAB

2024-10-31 21:18:29 ADT Last Updated: 2010-02-23	Richard Tervo [ tervo@unb.ca ]	Back to the course homepage...