Communications Toolbox

cyclpoly

Produce generator polynomials for a cyclic code

Syntax

• pol = cyclpoly(n,k);
  pol = cyclpoly(n,k,opt);
  

Description

For all syntaxes, a polynomial is represented as a row containing the coefficients in order of ascending powers.

pol = cyclpoly(n,k) returns the row vector representing one nontrivial generator polynomial for a cyclic code having codeword length n and message length k.

pol = cyclpoly(n,k,opt) searches for one or more nontrivial generator polynomials for cyclic codes having codeword length n and message length k. The output pol depends on the argument opt as shown in the table below.

opt	Significance of pol	Format of pol
'min'	One generator polynomial having the smallest possible weight	The row vector representing the polynomial
'max'	One generator polynomial having the greatest possible weight	The row vector representing the polynomial
'all'	All generator polynomials	A matrix, each row of which represents one such polynomial
a positive integer, L	All generator polynomials having weight L	A matrix, each row of which represents one such polynomial

The weight of a binary polynomial is the number of nonzero terms it has. If no generator polynomial satisfies the given conditions, then the output pol is empty and an error message is displayed.

Examples

The first command below produces representations of three generator polynomials for a [15,4] cyclic code. The second command shows that 1 + x + x² + x³+ x⁵+ x⁷+ x⁸+ x¹¹ is one such polynomial having the largest number of nonzero terms.

• c1 = cyclpoly(15,4,'all')
  c2 = cyclpoly(15,4,'max')
  

The output is

• c1 =
  
       1     1     0     0     0     1     1     0     0     0     1     1
       1     0     0     1     1     0     1     0     1     1     1     1
       1     1     1     1     0     1     0     1     1     0     0     1
  
  c2 =
  
       1     1     1     1     0     1     0     1     1     0     0     1
  

This command shows that no generator polynomial for a [15,4] cyclic code has exactly three nonzero terms.

• c3 = cyclpoly(15,4,3)
  
  No generator polynomial satisfies the given constraints.
  
  c3 =
  
       []
  

Algorithm

If opt is 'min', 'max', or omitted, then polynomials are constructed by converting decimal integers to base p. Based on the decimal ordering, gfprimfd returns the first polynomial it finds that satisfies the appropriate conditions. This algorithm is similar to the one used in gfprimfd.

See Also
cyclgen, encode

cyclgen

ddemod