Generator polynomial of BCH code


genpoly = bchgenpoly(n,k)
genpoly = bchgenpoly(n,k,prim_poly)
genpoly = bchgenpoly(n,k,prim_poly,outputFormat)
[genpoly,t] = bchgenpoly(...)


genpoly = bchgenpoly(n,k) returns the narrow-sense generator polynomial of a BCH code with codeword length n and message length k. The codeword length n must have the form 2m-1 for some integer m between 3 and 16. The output genpoly is a Galois row vector that represents the coefficients of the generator polynomial in order of descending powers. The narrow-sense generator polynomial is LCM[m_1(x), m_2(x), ..., m_2t(x)], where:

  • LCM represents the least common multiple,

  • m_i(x) represents the minimum polynomial corresponding to αi, α is a root of the default primitive polynomial for the field GF(n+1),

  • and t represents the error-correcting capability of the code.

    Note:   Although the bchgenpoly function performs intermediate computations in GF(n+1), the final polynomial has binary coefficients. The output from bchgenpoly is a Galois vector in GF(2) rather than in GF(n+1).

genpoly = bchgenpoly(n,k,prim_poly) is the same as the syntax above, except that prim_poly specifies the primitive polynomial for GF(n+1) that has Alpha as a root. prim_poly is an integer whose binary representation indicates the coefficients of the primitive polynomial in order of descending powers. To use the default primitive polynomial for GF(n+1), set prim_poly to [].

genpoly = bchgenpoly(n,k,prim_poly,outputFormat) is the same as the previous syntax, except that outputFormat specifies output data type. The value of outputFormat can be ‘gf' or 'double' corresponding to Galois field and double data types respectively. The default value of outputFormat is 'gf'.

[genpoly,t] = bchgenpoly(...) returns t, the error-correction capability of the code.


The results below show that a [15,11] BCH code can correct one error and has a generator polynomial X4 + X + 1.

m = 4;
n = 2^m-1; % Codeword length
k = 11; % Message length
% Get generator polynomial and error-correction capability.
[genpoly,t] = bchgenpoly(n,k)

The output is

genpoly = GF(2) array. 
Array elements = 
     1     0     0     1     1

t =



The maximum allowable value of n is 511.

More About


[1] Peterson, W. Wesley, and E. J. Weldon, Jr., Error-Correcting Codes, 2nd ed., Cambridge, MA, MIT Press, 1972.

See Also

| |

Was this topic helpful?