# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

# hammgen

Produce parity-check and generator matrices for Hamming code

## Syntax

h = hammgen(m)
h = hammgen(m,pol)
[h,g] = hammgen(...)
[h,g,n,k] = hammgen(...)

## Description

For all syntaxes, the codeword length is n. n has the form 2m – 1 for some positive integer m greater than or equal to 2. The message length, k, has the form nm.

h = hammgen(m) produces an m-by-n parity-check matrix for a Hamming code having codeword length n = 2^m-1. The input m is a positive integer greater than or equal to 2. The message length of the code is nm. The binary primitive polynomial used to produce the Hamming code is the default primitive polynomial for GF(2^m), represented by gfprimdf(m).

h = hammgen(m,pol) produces an m-by-n parity-check matrix for a Hamming code having codeword length n = 2^m-1. The input m is a positive integer greater than or equal to 2. The message length of the code is nm. pol is a row vector that gives the coefficients, in order of ascending powers, of the binary primitive polynomial for GF(2^m) that is used to produce the Hamming code. Alternatively, you can specify pol as a polynomial character vector. hammgen produces an error if pol represents a polynomial that is not, in fact, primitive.

[h,g] = hammgen(...) is the same as h = hammgen(...) except that it also produces the k-by-n generator matrix g that corresponds to the parity-check matrix h. k, the message length, equals n-m, or 2^m-1-m.

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

### Note

If your value of m is less than 25 and if your primitive polynomial is the default primitive polynomial for GF(2^m), the syntax hammgen(m) is likely to be faster than the syntax hammgen(m,pol).

## Examples

collapse all

Generate Hamming code matrices given codeword length.

Generate the parity check matrix h, the generator matrix g, the codeword length n, and the message length k for the Hamming code with .

[h,g,n,k] = hammgen(3)
h =

1     0     0     1     0     1     1
0     1     0     1     1     1     0
0     0     1     0     1     1     1

g =

1     1     0     1     0     0     0
0     1     1     0     1     0     0
1     1     1     0     0     1     0
1     0     1     0     0     0     1

n = 7
k = 4

Generate the parity check matrices for for the primitive polynomials and .

h1 = hammgen(4,'D^4+D+1');
h2 = hammgen(4,'D^4+D^3+1');

Remove the embedded 4-by-4 identity matrices (leftmost columns of both h1 and h2) and verify that the two matrices differ.

h1(:,5:end)
ans =

1     0     0     1     1     0     1     0     1     1     1
1     1     0     1     0     1     1     1     1     0     0
0     1     1     0     1     0     1     1     1     1     0
0     0     1     1     0     1     0     1     1     1     1

h2(:,5:end)
ans =

1     1     1     1     0     1     0     1     1     0     0
0     1     1     1     1     0     1     0     1     1     0
0     0     1     1     1     1     0     1     0     1     1
1     1     1     0     1     0     1     1     0     0     1

## Algorithms

Unlike gftuple, which processes one m-tuple at a time, hammgen generates the entire sequence from 0 to 2^m-1. The computation algorithm uses all previously computed values to produce the computation result.