Decoding by rank-2 bundles over plane quartics
Let \(X\) be a smooth curve of genus \(g\) defined over a finite field. \textit{T. Johnsen} [Int. J. Pure Appl. Math. 4, No. 1, 33--45 (2003; Zbl 1054.14031)] established a connection between Goppa codes based on divisors of \(X\) and rank-\(2\) bundles over \(X\). In the paper under review some steps are sketched towards a decoding algorithm based on the explicit construction of the rank-\(2\) bundle atached to a syndrome, as a \(2\)-extension of linear bundles. To this purpose it is necessary to detect subbundles in terms of given transition matrices of the rank-\(2\) bundle. Some examples are worked out for codes arising from Klein's quartic over a finite field of characteristic two.
scientific article; zbMATH DE number 5720156
scientific article