On decoding additive generalized twisted Gabidulin codes
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On decoding additive generalized twisted Gabidulin codes Wrya K. Kadir1 · Chunlei Li1 Received: 20 September 2019 / Accepted: 30 June 2020 / © The Author(s) 2020
Abstract In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field Fq n for any prime power q. This paper extends the work of the conference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded these codes over finite fields in characteristic two. Keywords Rank metric · Maximum rank distance codes · Gabidulin codes · Twisted Gabidulin codes · Generalized twisted Gabidulin codes Mathematics Subject Classification (2010) 94B35 · 68P30 · 11T71 · 11T06
1 Introduction Error correction codes with the rank metric have found applications in space-time coding [27], random network coding [44] and cryptography [12]. Many important properties of rank metric codes including the Singleton like bound were independently studied by Delsarte [9] , Gabidulin [13] and Roth [38]. Codes that achieve this bound were called maximum rank distance (MRD) codes. The most famous sub-family of MRD codes are Gabidulin codes which is the rank metric analog of Reed-Solomon codes. They have been extensively studied in the literature [9, 12, 13, 25, 36, 38]. Finding new families of MRD codes has been an interesting research topic since the invention of Gabidulin codes. In [20, 39], the Frobenious automorphism in the Gabidulin codes were generalized to arbitrary automorphism and generalized Gabidulin (GG) codes
Chunlei Li
[email protected] Wrya K. Kadir [email protected] 1
University of Bergen, Bergen, Norway
Cryptography and Communications
were proposed. In the past few years, a considerable amount of work has been done on MRD codes. In [40], Sheekey twisted the evaluation polynomial of a Gabidulin code and proposed a large family of MRD codes termed twisted Gabidulin (TG) codes. Using the same idea for generalizing Gabidulin codes, arbitrary automorphism was employed to construct generalized twisted Gabidulin (GTG) codes. This family of MRD codes were first described ¨ in [40, Remark 9] and later comprehensively studied in [26]. Otal and Ozbudak [30] later introduced a large family of MRD codes, known as additive generalized twisted Gabidulin (AGTG) codes, which contains all the aforementioned linear MRD codes as sub-families and new additive MRD codes. There are also some new families of MRD codes which are not equivalent to AGTG codes nor its subfamilies [5, 8, 42, 47]. Recent constructions of linear and nonlinear MRD codes were lately summarized in [31, 41]. MRD codes with efficient decoding algorithm are of great interest in practice. In his pioneering work [13], Gabidulin gave a decoding algorithm based on extended Euclidean algorithm. Subsequently, Richter and Plass in [36], and Loidreau [25] proposed modified version of Berlekamp-Massey and Welch-Berlekamp algorithms to deco
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