Алгеброгеометрические коды и декодирование на основе пар, исправляющих ошибки

Прикладная дискретная математика. 2023. № 62. С. 83–105.

Malygina E., Кунинец А. А., Раточка В. Л., Дупленко А. Г., Нейман Д. Я.

We consider the basic theory of algebraic curves and their function fields necessary for constructing algebraic geometry codes and a pair of codes constituting 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: алгеброгеометрический код, эллиптическая кривая, divisor, elliptic curve, дивизор, Function field, функциональное поле, algebraic-geometry code, error-correcting pair, decoding of algebraic-geometry code, Hermitian curve, Klein quartic, исправляющие ошибки пары, декодирование алгеброгеометрического кода, эрмитова кривая, квартика Клейна

Publication based on the results of: Research and development of data transmission and processing methods for the organization of energy efficient and secure data transmission in the Internet of Things networks (2024)