Performance analysis of a decoding algorithm for algebraic-geometry codes

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami-Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra, yielding similar asymptotic complexities.

Section: A Failure Probabilitysupporting

confidence: 58%

Section: Proofmentioning

confidence: 99%

Sub-Quadratic Decoding of One-Point Hermitian Codes

Nielsen

Beelen

2015

“…Under the BerlekampMassey-Sakata algorithm with majority voting, an error vector whose weight is larger than half the minimum distance of the code is often correctable. In particular this occurs for generic errors (also called independent errors in [16,17]), whose technical algebraic definition can be found in [18]. Generic errors of weight t can be a very large proportion of all possible errors of weight t, as in the case of the examples worked out in [15].…”

Section: Minimizing Redundancy For Correcting Generic Errorsmentioning

confidence: 99%

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Bras-Amorós¹,

O’Sullivan

2007

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding onepoint codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999.

“…Note that this condition is equivalent to det where Φ with det x l φ m = 0 is said to be generic, and [16], where such a Φ is said to be independent.…”

Section: Systematic Encoding Regarded As Erasure-only Decodingmentioning

confidence: 99%

Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

Matsui

2014

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Gröbner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n 3 ) using Gaussian elimination to O(qn 2 ) with some mild conditions on n and q, where n is the code length and q is the finite-field size.