Calculation of error-correcting pairs for an algebraicgeometric code | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2024. № 63. DOI: 10.17223/20710410/63/4

For an arbitrary algebrogeometric code and its dual, error-correcting pairs are explicitly calculated. Such a pair consists of codes that are necessary for an efficient decoding algorithm for a given code. The type of pairs depends on the powers of the divisors, with the help of which both the source code and one of the codes included in the pair are constructed. For an algebraic geometric code CL(D, G) of length and associated with a functional field F/Fq of genus g, pairs correcting t = ⌊(n - deg( G) - g - 1)/2⌋ errors, under certain restrictions on the degrees of divisors involved in their construction, are pairs of codes (Cl(D,F),Cl(D,G + F)⊥) or (CL(D,F⊥),Cl(D,F - G)). Restrictions on the degrees of divisors of codes (CL(D,F), CL(D,G - F)) constituting a pair correcting t = ⌊(deg( G) - 3g + 1)/2⌋ errors for the dual code CL(D, G)⊥) Cases where one of the codes involved in constructing a pair belongs to class of MDS codes and the parameters under which this situation is possible are derived. In addition, possible bounds for the divisors involved in the construction of error-correcting pairs for the subfield subcodes CL(D,G)|fp and CL< are calculated /sub>(D, G)⊥)|Fp of the original algebrogeometric code and its dual, with the expansion degree m = 2 (Fq = Fp2).