Further improvements on the designed minimum distance of algebraic geometry codes
Let \(F\) be an algebraic function field of genus \(g\) with full constant field \(\mathbb{F}_q\), where \(q\) is a prime power. Let \(G\) and \(D\) be two divisors of \(F\) with disjoint support and such that \(D\) is the sum of \(n\) different rational places. The residue Goppa code \(C_{\Omega}=C_{\Omega}(D,G)\) has length \(n\), dimension \(i(G-D)-i(G)\), and minimal distance \(d(C_{\Omega})\geq \deg G-(2g-2)\). There are a lot of results in the literature, improving this designed minimum distance. In the paper under review, the authors synthesize most of these results in two theorems that take the form: \[ d(C_{\Omega})- \deg G+(2g-2)\geq \deg Z+\left(i(A)-i(G-C)\right),\;\text{ or }\geq\deg Z+1, \] where \(A,C,Z\) are divisors satisfying opportune conditions. Most of the known improvements of the Goppa bound for the minimum distance are particular cases of these theorems. Also, some examples are discussed where these bounds improve on the previous ones. The ideas behind these improvements of the designed minimum distance are often based on \textit{shrinking} or \textit{growing} the divisors while keeping fixed their attached Riemann-Roch vector spaces. In this regard, the authors include in the paper a general analysis of the equivalence classes of divisors determined by the equivalence relation: \(M\approx N \Longleftrightarrow\mathcal{L}(M)=\mathcal{L}(N)\).
designed minimum distance, floor of a divisor
scientific article; zbMATH DE number 5364858
scientific article