Unique Decoding of General AG Codes

Abstract

A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed and regular structure with the previous algorithm. Compared with other known decoding algorithms for general AG codes, it is much simpler in its description and implementation.Comment: 11 pages,submitted to the IEEE Transactions on Information Theory; added a citation in the section III-

Decoding of differential AG codes

“…Let us define τ M (s) = (ν(s) − 1)/2 for s ∈Ω, which is the largest number of errors for which the majority voting succeeds for s. Like Proposition 22 in [8], we can show that for nongap s ≤ |G| − 2g + 2,…”

“…The bias is also reflected on the terms like primal AG code and dual AG code, which mean evaluation AG code and differential AG code respectively. In this respect, the recent result [8] on the unique decoding algorithm for evaluation AG codes is against to the trend and implies that the duality is not essential for decoding and for bounding the minimum distance of C L (D, G). This makes one conceive of a decoding algorithm for C Ω (D, G) that likewise does not rely on the duality, syndromes, or the space L(G).…”

Bounds for generalized Hamming weights of general AG codes

“…Then the evaluation AG code C L (G) is a [26,15,9] linear code over F 9 and the differential AG code C Ω (G) is a [26,11,13] linear code over F 9 . Note that these codes are examples in [13,14]. The following data were obtained using Magma [20].…”

“…Meanwhile the works [13] and [14] provided efficient and simple unique decoding algorithms for multi-point evaluation and differential AG codes, respectively. To construct the algorithms, the authors introduced the concept of Apéry systems defined for certain spaces of functions and differentials on the underlying curve.…”

On the evaluation codes given by simple $$\delta $$ δ -sequences

“…On the other hand, in [30] has been proved that AG codes can be decoded beyond the capacity of the algorithms previously mentioned. With a mix of this interpolation based list decoding and the syndrome decoding with majority voting scheme, it is shown in [37,38] how to decode certain family of one-point AG codes up to half of the AndersenGeil bound (see also [25,26,39]). These papers increase the interest on primary codes.…”