Constructions of block codes from algebraic curves over finite fields

Abstract

Abstract Since the discovery of algebraic geometry codes by Goppa in 1978–1982, mathematicians have been looking for other constructions of block codes from algebraic curves or varieties of higher dimension. In this survey article, we present various constructions of block codes from algebraic curves. As there is a one-to-one correspondence between algebraic curves and function fields, we adopt the function field language throughout this paper; this is advantageous for most of the constructions in this paper. 1 Introduction The discovery of algebraic geometry codes by Goppa has greatly stimulated research in both coding theory and number theory [1, 4, 5, 6, 7, 8, 9, 12, 16, 21, 24, 25, 33, 34, 44]. As there have been many other constructions of block codes via algebraic geometry, henceforth Goppa's algebraic geometry codes are called the Goppa geometric codes. The major breakthrough of the Goppa geometric codes is that they improved the long-standing benchmark bound, the Gilbert–Varshamov (GV, for short) bound. Before the Goppa geometric codes, the GV bound had remained for more than 30 years and many people even conjectured that the GV bound is optimal and could not be improved. The Goppa geometric codes are a natural generalization of the well-known Reed–Solomon codes from projective curves to algebraic curves of higher genus. Due to rich structures of algebraic function fields, there is a great potential to construct block codes via algebraic curves through other methods. Indeed, in the last few decades, various constructions of codes via algebraic curves have been found [2, 3, 8, 9, 17, 24, 26, 29, 37, 39, 40, 41, 42, 43, 44]. In this survey article, we present quite a few constructions of codes through algebraic curves. Many of these constructions have been discovered by the authors and their collaborators. The paper is organised as follows. In Section 2, we present some preliminaries on codes and algebraic curves. Section 3 is devoted to various constructions of codes.