An alternative decoding method for Gabidulin codes in characteristic zero

Abstract

Abstract-Gabidulin codes, originally defined over finite fields, are an important class of rank metric codes with various applications. Recently, their definition was generalized to certain fields of characteristic zero and a Welch-Berlekamp like algorithm with complexity O(n 3 ) was given. We propose a new application of Gabidulin codes over infinite fields: low-rank matrix recovery. Also, an alternative decoding approach is presented based on a Gao type key equation, reducing the complexity to at least O(n 2 ). This method immediately connects the decoding problem to well-studied problems, which have been investigated in terms of coefficient growth and numerical stability.

Reed–Solomon Codes over Fields of Characteristic Zero

“…Another important aspect is to study possible applications in detail. Moreover, the methods can be used to determine the bit complexity of the algorithms in [19], [20] for decoding Gabidulin codes over characteristic zero. Table I.…”

“…In this paper, we study RS codes over arbitrary fields and their properties. The idea is inspired by several recent publications that have dealt with Gabidulin codes, the rankmetric analog of RS codes, over field of characteristic zero (in particular number fields), which were first described in [17,Section 6], [18], [19], [20], [21]. These codes have applications in space-time coding [22] and low-rank matrix recovery [23].…”

Low-Rank Matrix Recovery using Gabidulin Codes in Characteristic Zero

“…This corresponds to the syndrome which is used by a decoder in order to produce the error matrix E which is in our case the matrix X 0 . We know from coding theory, that if rank(X 0 ) ≤ d−1 2 = n−k 2 = p 2n the matrix X 0 (and consequently E) is unique and can be found by a bounded minimum distance decoder [13]. ✷ Using our result from [13], decoding can be done in O(n 2 ).…”

“…We know from coding theory, that if rank(X 0 ) ≤ d−1 2 = n−k 2 = p 2n the matrix X 0 (and consequently E) is unique and can be found by a bounded minimum distance decoder [13]. ✷ Using our result from [13], decoding can be done in O(n 2 ). In comparison, the complexity of previous methods depends on singular value decomposition.…”

“…For decoding Gabidulin codes in characteristic zero, [11] gives a generalization of a Welch-Berlekamplike algorithm which allows decoding in O(n 3 ). In [13] we derived a Gao-like key equation and hence reduced the decoding complexity to O(n 2 ).…”