cyclpoly

Produce generator polynomials for 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.

optSignificance of polFormat of pol
'min'One generator polynomial having the smallest possible weight Row vector representing the polynomial
'max'One generator polynomial having the greatest possible weight Row vector representing the polynomial
'all'All generator polynomialsMatrix, each row of which represents one such polynomial
a positive integer, L All generator polynomials having weight L 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, the output pol is empty and a warning message is displayed.

Examples

collapse all

Create [15,4] cyclic code generator polynomials.

Use the input 'all' to show all possible generator polynomials for a [15,4] cyclic code. Use the input 'max' to show that $1+x+{\mathit{x}}^{2}+{x}^{3}+{x}^{5}+{x}^{7}+{x}^{8}+{x}^{11}$ is one such polynomial that has the largest number of nonzero terms.

c1 = cyclpoly(15,4,'all')
c1 = 3×12

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 = cyclpoly(15,4,'max')
c2 = 1×12

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)
Warning: No cyclic generator polynomial satisfies the given constraints.
c3 =

[]

Algorithms

If opt is 'min', 'max', or omitted, 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.