Step-By-Step Decoding of Alternant Codes

Abstract

In this paper we present a decoding scheme for Alternant codes. The syndromes are calculated from the received vector and the parity check matrix H. Let t be the error correcting capability of the decoder. Then we determine a Key Equation by adding t columns of the parity check matrix H. We raise this equation t-1 times to the power of n, where n is the number of columns of H. Next we consider a matrix At whose elements are the set of coefficients from the Key Equations which we obtained. We make a decision based on the determinant of the matrix A(t). If the matrix A(t) is singular, then we test the matrix A(t-1) for singularity and continue up to A(t-t+1) which in fact the decoder can correct one error. if any one of the matrices A(t) through A(t-t+1) is nonsingular we change the first digit of the received vector, then recompute the syndromes and recheck Δt'. If Δt' is zero the change is retained. If not, the digit is changed again. The Algorithm then proceeds to the next digit. This Algorithm for decoding Alternant codes has significant improvements over previous schemes since the step-by-step decoding can be carried out at selected areas of the received word.