Communications Toolbox

Logical Operations in Galois Fields

You can apply logical tests to Galois arrays and obtain a logical array. Some important types of tests are testing for equality of two Galois arrays and testing for nonzero values in a Galois array.

Testing for Equality

To compare corresponding elements of two Galois arrays that have the same size, use the operators == and ~=. The result is a logical array, each element of which indicates the truth or falsity of the corresponding elementwise comparison. If you use the same operators to compare a scalar with a Galois array, then MATLAB compares the scalar with each element of the array, producing a logical array of the same size.

• m = 5; r1 = gf([1:3],m); r2 = 1 ./ r1;
  lg1 = (r1 .* r2 == [1 1 1]) % Does each element equal one?
  lg2 = (r1 .* r2 == 1) % Same as above, using scalar expansion
  lg3 = (r1 ~= r2) % Does each element differ from its inverse?
  

The output is below.

• lg1 =
  
       1     1     1
  
  
  lg2 =
  
       1     1     1
  
  
  lg3 =
  
       0     1     1
  

Comparison of isequal and ==.   To compare entire arrays and obtain a logical scalar result rather than a logical array, you can use the built-in isequal function. Note, however, that isequal uses strict rules for its comparison, and returns a value of 0 (false) if you compare

• A Galois array with an ordinary MATLAB array, even if the values of the underlying array elements match
• A scalar with a nonscalar array, even if all elements in the array match the scalar

The example below illustrates this difference between == and isequal.

• m = 5; r1 = gf([1:3],m); r2 = 1 ./ r1;
  lg4 = isequal(r1 .* r2, [1 1 1]); % False
  lg5 = isequal(r1 .* r2, gf(1,m)); % False
  lg6 = isequal(r1 .* r2, gf([1 1 1],m)); % True
  

Testing for Nonzero Values

To test for nonzero values in a Galois vector, or in the columns of a Galois array that has more than one row, use the any or all function. These two functions behave just like the ordinary MATLAB functions any and all, except that they consider only the underlying array elements while ignoring information about which Galois field the elements are in. Examples are below.

• m = 3; randels = gf(randint(6,1,2^m),m);
  if all(randels) % If all elements are invertible
      invels = randels .\ 1; % Compute inverses of elements.
  else
      disp('At least one element was not invertible.');
  end
  alph = gf(2,4);
  poly = 1 + alph + alph^3;
  if any(poly) % If poly contains a nonzero value
      disp('alph is not a root of 1 + D + D^3.');
  end
  code = rsenc(gf([0:4;3:7],3),7,5); % Each row is a code word.
  if all(code,2) % Is each row entirely nonzero?
      disp('Both code words are entirely nonzero.');
  else
      disp('At least one code word contains a zero.');
  end
  

Arithmetic in Galois Fields

Matrix Manipulation in Galois Fields