Information theory and coding 2 final review BCH code

BCH code

BCH Construction of codes

For binary fields GF(2) And its extension GF(2^m), set up β=αi (i=1,2,…,2^m-2) by GF(2^m) Nonzero elements on , If GF(2) Polynomial on g(x) contain β,β²,…,β^d-1 etc. d-1 Continuous roots , By g(x) The generated cyclic code is called BCH code .d be called BCH Code design distance .

The original BCH Code and non primitive BCH code
If g(x) Of d-1 Continuous roots contain primitive elements , said g(x) Generated BCH Code is primitive BCH code ;
If g(x) Of d-1 Every continuous root is a non primitive , be g(x) Generated BCH Codes are called non primitives BCH code .

Generate polynomial and code length

The original BCH Construction steps of code

1. According to yard length n=2^m-1 determine m, Look up the table to find out m Subprimitive polynomial p(x), Construct extended domain GF(2^m)
2. Take the primitive α, According to the design error correction capability t determine g(x) The root of the ： α,α², α³,…,α^2t, Look up the table to find the minimum polynomial of the root $M_1$(x), $M_3$(x), …,$M_{(}2t-1)$(x)
3. Calculate the minimum common multiple of the above minimum polynomial , Get the generating polynomial g(x).

Example ：
To design error correction capability t=1,2,3 Construct code length separately n=15 The origin of BCH code .

BCH Check matrix of code