Open AccessProperties of triple error orbits G and their invariants in Bose – Chaudhuri – Hocquenghem codes C7
Author(s) -
V. A. Lipnitski,
A. U. Serada
Publication year - 2019
Publication title -
vescì nacyânalʹnaj akadèmìì navuk belarusì. seryâ fìzìka-tèhnìčnyh navuk
Language(s) - English
Resource type - Journals
eISSN - 2524-244X
pISSN - 1561-8358
DOI - 10.29235/1561-8358-2019-64-1-110-117
Subject(s) - bch code , mathematics , scalar (mathematics) , automorphism , polynomial , gröbner basis , discrete mathematics , group (periodic table) , combinatorics , pure mathematics , algebra over a field , decoding methods , algorithm , physics , mathematical analysis , geometry , quantum mechanics
This work is the further development of the theory of norms of syndromes: the theory of polynomial invariants of G-orbits of errors expands with the group G of automorphisms of binary cyclic BCH codes obtained by joining the degrees of cyclotomic permutation to the group Γ and practically exhausting the group of automorphisms of BCH codes. It is determined that polynomial invariants, like the norms of syndromes, have a scalar character and are one-to-one characteristics of their orbits for BCH codes with a constructive distance of five. The paper introduces the corresponding vector polynomial invariants for primitive cyclic BCH codes with a constructive distance of seven, next to the norms of the syndromes that are already vector quantities; the basic properties of the vector polynomial invariants are investigated. It is established that the property of mutual unambiguity is violated: there are G-orbit-isomers, which are different, but have the same vector polynomial invariants. It is substantiated and demonstrated by examples that this circumstance greatly complicates error decoding algorithms based on polynomial invariants