Reduced Row Echelon Matrix Form over GF(q^k)

This function takes a matrix A defined over GF(q^k), where the elements of A are in exponential form, and outputs the matrix in RREF.

%% Description:

% This function converts matrix A to Reduced Row Echelon Form

% (RREF) using arithmetic over GF(q^k), where A contains

% entries over GF(q^k) and A is in exponential form. This means

% that if A(1,1) is 1, then this corresponds to the exponent of

% β^1 where β is a primitive element of GF(q^k). In this

% function, -inf corresponds to the 0 element of GF(q^k), since

% 0 cannot be expressed in exponential form.

%

% Inputs:

% A: Matrix in exponential form over GF(q^k)

% q: base field parameter (should be prime)

% k: extension field parameter

%

% Outputs:

% R: Matrix in RREF form, expressed exponentially over GF(q^k)

% field: Outputs a GF(q^k) table. Exponential form can be

% mapped to polynomial form using this table. For example,

% field(i,:) is the polynomial form of β^(i-2) in order of

% lowest order to highest order. Note that field(1,:)

% corresponds to the 0 element of GF(q^k).

%

% Dependencies:

%

% function gfksub: This function performs a-b over GF(q^k)

% arithmetic, where a,b are elements GF(q^k). This function is

% included in this file.

%

% References:

% RREF algorithm: Yuanzhen Shao- AN ALGORITHM FOR REDUCING A MATRIX TO

% ROW ECHELON FORM

% https://www.math.purdue.edu/~shao92/documents/Algorithm%20REF.pdf

%

% Credits:

% Author: Jacqueline Malayter

% Date: July 18, 2025

%

% This code was developed by Jacqueline Malayter. If you use or adapt

% this code in your own work or publication, please credit the

% original author by including a reference in your code comments or

% documentation.

%

% License: This work is licensed under the Creative Commons Attribution

% 4.0 International License (CC BY 4.0). To view a copy of this

% license, visit: https://creativecommons.org/licenses/by/4.0/