gf

Galois field array

Syntax

x_gf = gf(x)

x_gf = gf(x,m)

x_gf = gf(x,m,prim_poly)

Description

x_gf = gf(x) creates a Galois field (GF) array, GF(2), from matrix x.

x_gf = gf(x,m) creates a Galois field array from matrix x. The Galois field has 2^m elements, where m is an integer from 1 through 16.

x_gf = gf(x,m,prim_poly) creates a Galois field array from matrix x by using the primitive polynomial prim_poly.

Examples

Create GF(2) Array from Specified Matrix

Specify a matrix of 0s and 1s.

x = [0 1 1; 0 1 0; 1 1 1];

Create a GF(2) array from x.

x_gf = gf(x)

 
x_gf = GF(2) array. 
 
Array elements = 
 
   0   1   1
   0   1   0
   1   1   1

Create Sequence of GF(16) Elements

Set the order of the Galois field to 16, where the order equals $$2^m$$. Specify a matrix of elements that range from 0 to $$2^m-1$$. Create the Galois field array.

m = 4;
x = [3 2 9; 1 2 1];
y = gf(x,m)

 
y = GF(2^4) array. Primitive polynomial = D^4+D+1 (19 decimal)
 
Array elements = 
 
   3   2   9
   1   2   1

Create GF Sequence with Specified Primitive Polynomial

Create a sequence of integers. Create a Galois field array in GF($2^5$).

x = [17 8 11 27];
y = gf(x,5)

 
y = GF(2^5) array. Primitive polynomial = D^5+D^2+1 (37 decimal)
 
Array elements = 
 
   17    8   11   27

Determine all possible primitive polynomials for GF($2^5$).

pp = primpoly(5,'all')

 
Primitive polynomial(s) = 
 
D^5+D^2+1
D^5+D^3+1
D^5+D^3+D^2+D^1+1
D^5+D^4+D^2+D^1+1
D^5+D^4+D^3+D^1+1
D^5+D^4+D^3+D^2+1

pp = 6×1

    37
    41
    47
    55
    59
    61

Create a Galois field array using the primitive polynomial that has a decimal equivalent of 59.

z = gf(x,5,'D5+D4+D3+D+1')

 
z = GF(2^5) array. Primitive polynomial = D^5+D^4+D^3+D+1 (59 decimal)
 
Array elements = 
 
   17    8   11   27

Input Arguments

x — Input matrix

matrix with all values greater than or equal to zero

Input matrix, specified as a matrix with values greater than or equal to zero. The function uses this value to create a GF array.

• If you do not specify the prim_poly input argument, each element of x must be an integer in the range [0, 2^m–1].

• If you specify prim_poly input argument, each element of x must be 0 or 1.

Data Types: double

m — Order of primitive polynomial

positive integer

Order of primitive polynomial, specified as a positive integer from 1 through 16. The function uses this value to calculate the distinct number of elements in the GF.

Data Types: double

prim_poly — Primitive polynomial

primitive polynomial in GF(2^m) (default) | binary row vector | character vector | string scalar | positive integer

Primitive polynomial, specified as one of these options:

• Binary row vector — This vector specifies coefficients of prim_poly in the order of ascending powers.

• Character vector or a string scalar — This value defines prim_poly in a textual representation. For more details, refer to polynomial character vector.

• Positive integer — This value defines prim_poly in the range [(2^m + 1), (2^m+1 – 1)].

If prim_poly is not specified, see Default Primitive Polynomials for the list of default primitive polynomial used for each Galois field array GF(2^m).

Data Types: double | char | string

Output Arguments

x_gf — Galois field array

variable that MATLAB recognizes as a Galois field array

Galois field array, returned as a variable that MATLAB recognizes as a Galois field array, rather than an array of integers. As a result, when you manipulate the variable, MATLAB works within the Galois field the variable specifies. For example, if you apply the log function to a Galois array, MATLAB computes the logarithm in the Galois field for that Galois array and not in the field of real or complex numbers.

More About

Default Primitive Polynomials

This table lists the default primitive polynomial used for each Galois field array GF(2^m). To use a different primitive polynomial, specify prim_poly as an input argument. prim_poly must be in the range [(2^m + 1), (2^m+1 – 1)] and must indicate an irreducible polynomial. For more information, see Primitive Polynomials and Element Representations.

Value of m	Default Primitive Polynomial	Integer Representation
1	D + 1	3
2	D2 + D + 1	7
3	D3 + D + 1	11
4	D4 + D + 1	19
5	D5 + D2 + 1	37
6	D6 + D + 1	67
7	D7 + D3 + 1	137
8	D8 + D4 + D3 + D2 + 1	285
9	D9 + D4 + 1	529
10	D10 + D3 + 1	1033
11	D11 + D2 + 1	2053
12	D12 + D6 + D4 + D + 1	4179
13	D13 + D4 + D3 + D + 1	8219
14	D14 + D10 + D6 + D + 1	17475
15	D15 + D + 1	32771
16	D16 + D12 + D3 + D + 1	69643

Galois Computations

This table lists the operations supported for Galois field arrays.

Operation	Description
+ -	Addition and subtraction of Galois arrays
* / \	Matrix multiplication and division of Galois arrays
.* ./ .\	Elementwise multiplication and division of Galois arrays
^	Matrix exponentiation of Galois array
.^	Elementwise exponentiation of Galois array
' .'	Transpose of Galois array
==, ~=	Relational operators for Galois arrays
all	True if all elements of a Galois vector are nonzero
any	True if any element of a Galois vector is nonzero
conv	Convolution of Galois vectors
convmtx	Convolution matrix of Galois field vector
deconv	Deconvolution and polynomial division
det	Determinant of square Galois matrix
dftmtx	Discrete Fourier transform matrix in a Galois field
diag	Diagonal Galois matrices and diagonals of a Galois matrix
fft	Discrete Fourier transform
filter (gf)	One-dimensional digital filter over a Galois field
ifft	Inverse discrete Fourier transform
inv	Inverse of Galois matrix
length	Length of Galois vector
log	Logarithm in a Galois field
lu	Lower-Upper triangular factorization of Galois array
minpol	Find the minimal polynomial for a Galois element
mldivide	Matrix left division \ of Galois arrays
polyval	Evaluate polynomial in Galois field
rank	Rank of a Galois array
reshape	Reshape Galois array
roots	Find polynomial roots across a Galois field
size	Size of Galois array
tril	Extract lower triangular part of Galois array
triu	Extract upper triangular part of Galois array