根据H在有限域GF(2^m)上求解生成矩阵G

原理

有时间再补充。

注1：使用高斯消去法。如果Py不为单位阵，则说明进行了列置换，此时G不是系统形式。

注2：校验矩阵H必须是行满秩才存在对应的生成矩阵G，且生成矩阵G通常不唯一。

matlab实现：只做列置换，不做行置换

function [G, Px, Py] = Gaussian_Elimination_in_GFq(H, q)
    
    % initial
    [m, n] = size(H);
    H_stair = mod(H,q);
    Px = eye(m); 
    Py = eye(n);
    
    for i=1:m
        for j=i:n
            if gcd(H_stair(i,j), q) == 1
                break;
            else
                if j==n
                    error('Gaussian_Elimination_in_GFq: The H is not full rank in GF(%d).',q);
                else
                    continue;
                end
            end
        end

        if j ~= i
            H_stair(:, [i, j]) = H_stair(:, [j, i]);
            Py(:, [i, j]) = Py(:, [j, i]);
        end
    end
    
    for i=1:m
        [~, x, ~] = gcd(H_stair(i,i), q);
        inv_i = mod(x, q);

        H_stair(i, :) = mod(inv_i * H_stair(i, :), q);
        Px(i, :) = mod(inv_i * Px(i, :), q);

        for j=1:m
            if j~=i && H_stair(j,i)~=0
                factor=H_stair(j, i);
                H_stair(j, :) = mod(H_stair(j, :) - factor * H_stair(i, :), q);
                Px(j, :) = mod(Px(j, :) - factor * Px(i, :), q);
            end
        end
    end

    parity_matrix = mod((q-1).*H_stair(:,m+1:n),q);
    G_fixed=[parity_matrix',eye(n-m)];
    G=G_fixed*Py';
end

matlab实现：先做行置换，若无逆元，再做列置换

function [G, Px, Py] = Gaussian_Elimination_in_GFq(H, q)
    
    [m, n] = size(H);
    H_stair = mod(H,q);
    Px = eye(m); 
    Py = eye(n);
    
    %% column find
    for j=1:m
        for i=j:m
            if gcd(H_stair(i,j), q) == 1
                break;
            else
                if i==m
                    %% row find
                    for k=j+1:n
                        if gcd(H_stair(j,k), q) == 1
                            break;
                        else
                            if k==n
                                error('Gaussian_Elimination_in_GFq: The H is not full rank in GF(%d).',q);
                            else
                                continue;
                            end
                        end
                    end
                    %% column exchange
                    H_stair(:,[j,k]) = H_stair(:,[k,j]);
                    Py(:,[j,k]) = Py(:,[k,j]);
                    i=j;
                    break;
                else
                    continue;
                end
            end
        end
        %% row exchange
        if i ~= j
            H_stair([j, i],:) = H_stair([i, j],:);
            Px([j, i],:) = Px([i, j],:);
        end
    end

    for i=1:m
        [~, x, ~] = gcd(H_stair(i,i), q);
        inv_i = mod(x, q);

        H_stair(i, :) = mod(inv_i * H_stair(i, :), q);
        Px(i, :) = mod(inv_i * Px(i, :), q);

        for j=1:m
            if j~=i && H_stair(j,i)~=0
                factor=H_stair(j, i);
                H_stair(j, :) = mod(H_stair(j, :) - factor * H_stair(i, :), q);
                Px(j, :) = mod(Px(j, :) - factor * Px(i, :), q);
            end
        end
    end

    parity_matrix = mod((q-1).*H_stair(:,m+1:n),q);
    G_fixed=[parity_matrix',eye(n-m)];
    G=G_fixed*Py';
    
end