Function fields and generalized product codes

Product codes are generalized by using generator polynomials in two variables (g₁(X,Y), g ₂(X,Z), etc.) and by interpreting them as polynomials in one variable with coefficients belonging to the field F(X) of rational functions in the variable X over a finite field F. With such an interpretation the polynomial g₁(X,Y) of degree d in Y defines an algebraic function y=Y( X) of X which generates an algebraic function field K of degree d over F(X). Such function fields are the basis for the decoding algorithm in which error polynomials E_j(X) are determined by use of linear equations with coefficients belonging to K. One such equation corresponds to d equations over F(X), which results in inversion of matrices of order d with elements which are polynomials in X. It is shown how to use subfields in an algebraic function field of degree d=a×b to increase the dimension of a code and at the same time replace a generator polynomial g₁(X,Y) of degree d in Y with two generator polynomials _g2(X,Y ), g₃(X,Z) of degrees a and b, respectively