Documentation

# 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`.