Communications Toolbox

cyclgen

Produce parity-check and generator matrices for cyclic code

Syntax

• h = cyclgen(n,pol);
  h = cyclgen(n,pol,opt);
  [h,g] = cyclgen(...);
  [h,g,k] = cyclgen(...);
  

Description

For all syntaxes, the codeword length is n and the message length is k. A polynomial can generate a cyclic code with codeword length n and message length k if and only if the polynomial is a degree-(n-k) divisor of x^n-1. (Over the binary field GF(2), x^n-1 is the same as x^n+1.) This implies that k equals n minus the degree of the generator polynomial.

h = cyclgen(n,pol) produces an (n-k)-by-n parity-check matrix for a systematic binary cyclic code having codeword length n. The row vector pol gives the binary coefficients, in order of ascending powers, of the degree-(n-k) generator polynomial.

h = cyclgen(n,pol,opt) is the same as the syntax above, except that the argument opt determines whether the matrix should be associated with a systematic or nonsystematic code. The values for opt are 'system' and 'nonsys'.

[h,g] = cyclgen(...) is the same as h = cyclgen(...) except that it also produces the k-by-n generator matrix g that corresponds to the parity-check matrix h.

[h,g,k] = cyclgen(...) is the same as [h,g] = cyclgen(...) except that it also returns the message length k.

Examples

The code below produces parity-check and generator matrices for a binary cyclic code with codeword length 7 and message length 4.

• pol = cyclpoly(7,4);
  [parmat,genmat,k] = cyclgen(7,pol)
  

The output is

• parmat =
  
       1     0     0     1     1     1     0
       0     1     0     0     1     1     1
       0     0     1     1     1     0     1
  
  
  genmat =
  
       1     0     1     1     0     0     0
       1     1     1     0     1     0     0
       1     1     0     0     0     1     0
       0     1     1     0     0     0     1
  
  
  k =
  
       4
  

In the output below, notice that the parity-check matrix is different from parmat above, because it corresponds to a nonsystematic cyclic code. In particular, parmatn does not have a 3-by-3 identity matrix in its leftmost three columns, as parmat does.

• parmatn = cyclgen(7,cyclpoly(7,4),'nonsys')
  
  parmatn =
  
       1     1     1     0     1     0     0
       0     1     1     1     0     1     0
       0     0     1     1     1     0     1
  

See Also
encode, decode, bchpoly, cyclpoly

cosets

cyclpoly