The BMS algorithm and decoding of AG codes

Abstract

In this paper, we review various decoding methods of algebraic geometry (or algebraic-geometric) codes (Goppa in Soviet Math. Dokl. 24(1):170-172, 1981; Hoholdt et al. in Handbook of coding theory, vols. I, II, North-Holland, Amsterdam, pp. 871-961, 1998; Geil in Algebraic geometry codes from order domains, this volume, pp. 121-141, 2009) mainly based on the Gröbner basis theory (Buchberger in Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965; Aequationes Math. 4:374-383, 1970; Multidimensional systems theory, Reidel, Dordrecht, pp. 184-232, 1985; London Math. Soc. LNS 251:535-545, 1998; J. Symb. Comput. 41(3-4):475-511, 2006; Mora in Gröbner technology, this volume, pp. 11-25, 2009b) as well as the BMS algorithm (Sakata in J. Symbolic Comput. 5(3):321-337, 1988; Inform. and Comput. 84(2):207-239, 1990) and its variations (Sakata in n-dimensional Berlekamp-Massey algorithm for multiple arrays and construction of multivariate polynomials with preassigned zeros, LNCS, vol. 357, pp. 356-376, 1989; Finding a minimal polynomial vector set of a vector of nD arrays, LNCS, vol. 539, pp. 414-425, 1991), where the BMS algorithm itself is reviewed in another paper (Sakata in The BMS algorithm, this volume, pp. 143-163, 2009) in this issue. The main subjects are: (1) Syndrome decoding of dual codes up to the designed distance (Saints and Heegard in IEEE Trans. Inform. Theory 41(6):1733-1751, 1995; Sakata et al. in Finite Fields Appl. 1(1):83-101, 1995b; IEEE Trans. on Inf. Th. 41(6):1672-1677, 1995c; IEEE Trans. on Inf. Th. 41(6):1762-1768, 1995a) by using the BMS algorithm. (There have been published several methods of decoding algebraic geometry codes, e.g. Kötter in On decoding of algebraic-geometric and cyclic codes, Ph.D. thesis, Linköping University, 1996; O'Sullivan in IEEE Trans. on Inf. Th. 41(6):1709-1719, 1995; Guerrini and Rimoldi in FGLM-like decoding: from Fitzpatrick's approach to recent developments, this volume, pp. 197-218, 2009, which are described in some terminology rather from the perspective of algebraic geometry, but are in principle equivalent to the BMS decoding method. We omit their descriptions here.) (2) List decoding of primal codes (Numakami et al. in IEICE Trans. Fundamentals J83:1309-1317, 2000; Sakata in LNCS, vol. 2227, pp. 172-181, 2001; Proc. of ISIT2003, pp. 363-363, 2003). (The original list decoding algorithms are given for RS codes by Sudan in J. of Complexity 13:180-193, 1997, and for algebraic geometry codes by Shokrollahi and Wassermann in IEEE Trans. on Inf. Th. 45(2):432-437, 1999, and their improved versions by Guruswami and Sudan in IEEE Trans. on Inf. Th. 45:(6):1757-1767, 1999.) (3) Other relevant decoding algorithms of primal and dual codes (Augot in Proc. of ISIT2002, pp. 86-86, 2002; Justesen and Hoholdt in A course in error-correcting codes, EMS Textbooks in Mathematics, EMS, 2004; Fujisawa and Sakata in Proc. of SITA2005, pp. 543-546, 2005; Sakata and Fujisawa in Proc. of SITA2006, pp. 93-96, 2006; Fujisawa et al. in Proc. of SITA2006, pp. 101-104, 2006). In discussing list decoding and usual bounded-distance decoding of primal/dual codes we show that multi-variate interpolation problem is a key and that it can be solved by using the BMS algorithm efficiently. The computational complexities of our methods are less than the other decoding methods including the Feng-Rao (IEEE Trans. on Inf. Th. 39(1):37-45, 1993) algorithm simply based on Gaussian elimination. These reductions in computational complexity are based on the special structures or properties of the given input data (syndrome arrays, etc.) which originate in the definition of codes themselves and are used cleverly by the BMS algorithm. In Leonard (A tutorial on AG code decoding from a Gröbner basis perspective, this volume, pp. 187-196, 2009b), Guerrini and Rimoldi (FGLM-like decoding: from Fitzpatrick's approach to recent developments, this volume, pp. 197-218, 2009) in this issue, several other efficient decoding methods of algebraic geometry codes from Gröbner basis perspectives are reviewed. Additionally, we mention a recent development of decoding algorithm based on higher-dimensional interpolation (Parvaresh and Vardy in Proc. of IEEE FOCS2005, IEEE Computer Society, pp. 285-294, 2005), which has error correction performance superior to the improved list decoding by Guruswami and Sudan. As a general method of multivariate interpolation the BMS algorithm is an alternative of the Buchberger-Möller (The construction of multivariate polynomials with preassigned zeros, LNCS, vol. 144, pp. 24-31, 1982), Mora (The FGLM problem and Möller's algorithm on zero-dimensional ideals, this volume, pp. 27-45, 2009a) algorithm and the Marinani-Möller-Mora (AAECC 4:(2):103-145, 1993) algorithm, but any exact comparisons of computational complexities of these methods remain to be investigated. © 2009 Springer-Verlag Berlin Heidelberg.