Sub-Quadratic Decoding of One-Point Hermitian Codes

Abstract

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.

Improved power decoding of interleaved one-point Hermitian codes

“…The proof is similar to [4,Lemma 23]. Uniqueness is clear since if there were two such polynomials, their difference would also be in L − i∈E sP i + ∞P∞ , but of smaller deg H .…”

“…Being in L − i∈E sP i + ∞P∞ specifies s|E| homogeneous linear equations in the coefficients of Λs, since for any i ∈ E, we can expand Λs into a power series j≥s γ i,j φ j i for a local parameter φ i of P i (e.g., take φ i = X − α i if P i = (α i , β i )). By requiring deg H Λs ≤ s|E| + g, we have more variables than equations, so there is a non-zero Λs of the sought form with degree at most s|E| + g. The lower bound works exactly as in [4,Lemma 23].…”

“…Let q be a prime power. We follow the notation of [4]. The Hermitian curve H/F q 2 is the smooth projective plane curve defined by the affine equation Y q + Y = X q+1 .…”

“…In this section, we derive the system of key equations that we need for decoding, using the same trick as [12] for interleaved Reed-Solomon codes. We use the description of power decoding for one-point Hermitian codes as in [4]. Suppose that the received word is r = c + e ∈ F h×n q 2 , consisting of an error e with corresponding (burst) error positions E and a codeword c ∈ C H (n, m H ; h), which is obtained from the message polynomials…”

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

“…• Minimal approximant bases [15,53] were used to compute kernel bases [54], giving the first efficient deterministic algorithm for linear system solving over K [x]. • Basis reduction [15,16] played a key role in accelerating the decoding of one-point Hermitian codes [35] and in designing deterministic determinant and Hermite form algorithms [29]. • Progress on minimal interpolant bases [23,24] led to the best known complexity bound for list-decoding Reed-Solomon codes and folded Reed-Solomon codes [24, Sec.…”