Communications Toolbox

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 2^m-1 for some positive integer m greater than or equal to 3. The message length, k, has the form n-m.

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 3. The message length of the code is n-m. 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 3. The message length of the code is n-m. 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. 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), then the syntax hammgen(m) is likely to be faster than the syntax hammgen(m,pol).

Examples

The command below exhibits the parity-check and generator matrices for a Hamming code with codeword length 7 = 2³-1 and message length 4 = 7-3.

• [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
  

The command below, which uses 1 + x² + x³ as the primitive polynomial for GF(2³), shows that the parity-check matrix depends on the choice of primitive polynomial. Notice that h1 below is different from h in the example above.

• h1 = hammgen(3,[1 0 1 1])
  
  h1 =
  
       1     0     0     1     1     1     0
       0     1     0     0     1     1     1
       0     0     1     1     1     0     1
  

Algorithm

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.

See Also
gftuple, gfrepcov, gfprimck, gfprimfd, gfprimdf

gfweight

hank2sys