Communications Toolbox

Matrix Manipulation in Galois Fields

Some basic operations that you would perform on an ordinary MATLAB array are available for Galois arrays. This section illustrates how to perform basic manipulations and how to get basic information.

Basic Manipulations of Galois Arrays

Basic array operations that you can perform on a Galois array include those in the table below. The results of these operations are Galois arrays in the same field. The functionality of these operations is analogous to the MATLAB operations having the same syntax.

Operation	Syntax
Index into array, possibly using colon operator instead of a vector of explicit indices	a(vector) or a(vector,vector1), where vector and/or vector1 can be ":" instead of a vector
Transpose array	a'
Concatenate matrices	[a,b] or [a;b]
Create array having specified diagonal elements	diag(vector) or diag(vector,k)
Extract diagonal elements	diag(a) or diag(a,k)
Extract lower triangular part	tril(a) or tril(a,k)
Extract upper triangular part	triu(a) or triu(a,k)
Change shape of array	reshape(a,k1,k2)

The code below uses some of these syntaxes.

• m = 4; a = gf([0:15],m);
  a(1:2) = [13 13]; % Replace some elements of the vector a.
  b = reshape(a,2,8); % Create 2-by-8 matrix.
  c = [b([1 1 2],1:3); a(4:6)]; % Create 4-by-3 matrix.
  d = [c, a(1:4)']; % Create 4-by-4 matrix.
  dvec = diag(d); % Extract main diagonal of d.
  dmat = diag(a(5:9)); % Create 5-by-5 diagonal matrix
  dtril = tril(d); % Extract upper and lower triangular
  dtriu = triu(d); % parts of d.
  

Basic Information About Galois Arrays

You can determine the length of a Galois vector or the size of any Galois array using the length and size functions. The functionality for Galois arrays is analogous to that of the MATLAB operations on ordinary arrays, except that the output arguments from size and length are always integers, not Galois arrays. The code below illustrates the use of these functions.

• m = 4; e = gf([0:5],m); f = reshape(e,2,3);
  lne = length(e); % Vector length of e
  szf = size(f); % Size of f, returned as a two-element row 
  [nr,nc] = size(f); % Size of f, returned as two scalars
  nc2 = size(f,2); % Another way to compute number of columns
  

Positions of Nonzero Elements.   Another type of information you might want to determine from a Galois array is the positions of nonzero elements. For an ordinary MATLAB array, you might use the find function. However, for a Galois array you should use find in conjunction with the ~= operator, as illustrated.

• x = [0 1 2 1 0 2]; m = 2; g = gf(x,m);
  nzx = find(x); % Find nonzero values in the ordinary array x.
  nzg = find(g~=0); % Find nonzero values in the Galois array g.
  

Logical Operations in Galois Fields

Linear Algebra in Galois Fields