On Erasure Decoding of AG-codes

We present a scheme for erasure decoding of AG-codes of complexity O(n2). This improves on methods involving Fourier transforms. The trivial scheme for erasure decoding of AG-codes (Algebraic Geometry codes in full, or Geometric Goppa codes) solves a system of linear equations. The scheme is of complexity O(n), where n denotes the codelength. Fast schemes, that use a Fourier transform of all syndromes, improve on the complexity. For codes over GF (q) defined with curves in r-dimensional affine space, there are (q − 1) syndromes to be computed. The transformation itself is of complexity O(nq) [1]. In general, q > n. The determination of syndromes is computationally equivalent to the determination of the message symbols. We give a scheme for the computation of the message symbols. It avoids the computation of the syndromes and the use of the Fourier transform. The scheme requires 3kn field multiplications, with k the dimension of the code. Notation 1 The notation will be as follows. Let C be a linear code of type [n, k], with generator matrix G and parity check matrix H. Let m = (mi), c = (ci), e = (ei),y = (yi) denote a message, the encoded message, an error vector, and the received message respectively. The vectors are related via c = mG, y = c+ e. Let it be known that errors occurred only at the coordinates I ⊂ {1, 2, . . . , n}. For a fixed code C, erasure decoding concerns the determination of the message m, given the received message y and the set of unreliable positions I. In algebraic decoding, the set of unreliable positions I will in general be given as the set of zeros of an error-locating vector [2]. Definition 2 By u∗e == (uiei) we denote the componentwise product of vectors u and e. A vector u is called error-locating if it has the support of e among its zeros. Equivalently, if u ∗ e = 0. We recall two known solutions to erasure decoding. Both compute the error vector first. Proposition 3 The error vector e can be computed from the system of linear equations (1.1) He = Hy , ei = 0, for i 6∈ I. Alternatively, for a regular square matrix H̄ of size n′ ≥ n, and a complete syndrome vector s = H̄e , it can be obtained as (1.2) e = H̄−1sT . The computation of the message vector then proceeds in both cases via (2) Compute c = y − e. (3) Compute m, with c =mG.