A Unified Algorithm for Solving Key Equations for Decoding Alternant Codes

for given non-zero polynomials G,Ψ, S. A polynomial trio (σ, η, ω) satisfying (1) may be referred to as a minimal solution of the key equation if σ is monic; deg(σΨ) > deg(ω); and deg(σ) is minimum. Let A(G,Ψ, S) denote the set of all minimal solutions of key equation (1). The problem of finding, from a known element of A(G,Ψ, S), one element each of: (a) A(G,Ψ/(x− β1), S); (b) A(G, (x− β2)Ψ, S); (c) A((x− β3)G,Ψ, S); (d) A(G/(x− β4),Ψ, S), (where βi ∈ F for 1 ≤ i ≤ 4, Ψ(β1) = 0, and G(β4) = 0) is the problem we address in this paper. This problem may be thought of as an extension of the problem, discussed by Berlekamp [2] and Massey [14], of finding an element of A(xm+1,Ψ, S) given that an element of A(xm,Ψ, S) has been found. In this paper, we present the solution to this problem by an efficient algorithm derived on the basis of the Berlekamp-Massey algorithm [2], [14] and the algorithms presented in [4],[8],[10],[12],[19],[20] . We show that this algorithm can be applied to a number of decoding techniques, including boundeddistance (BD) decoding, generalized minimum distance