This article is cited in 2 papers

Applied Coding Theory

Algebraic-geometry codes and decoding by error-correcting pairs

E. S. Malygina^a, A. A. Kuninets^b, V. L. Ratochka^b, A. G. Duplenko^b, D. Y. Neyman<sup>b

a</sup> HSE, Moscow, Russia
^b Immanuel Kant Baltic Federal University, Kaliningrad, Russia

Abstract: We consider the basic theory of algebraic curves and their function fields necessary for constructing algebraic geometry codes and a pair of codes forming an error-correction pair which is used in a precomputation step of the decoding algorithm for the algebraic geometry codes. Also, we consider the decoding algorithm and give the necessary theory to prove its correctness. As a result, we consider elliptic curves, Hermitian curves and Klein quartics and construct the algebraic geometry codes associated with these families of curves, and also explicitly define the error-correcting pairs for the resulting codes.

Keywords: algebraic geometry code, function field, divisor, error-correcting pair, decoding of algebraic geometry code, elliptic curve, Hermitian curve, Klein quartic.

UDC: 519.725

DOI: 10.17223/20710410/62/7