On the evaluation codes given by simple $$\delta $$ δ -sequences

Abstract

Abstract. Plane valuations at infinity are classified in five types. Valuations in one of them determine weight functions which take values on semigroups of Z 2 . These semigroups are generated by δ-sequences in Z 2 . We introduce simple δ-sequences in Z 2 and study the evaluation codes of maximal length that they define. These codes are geometric and come from order domains. We give a bound on their minimum distance which improves the Andersen-Geil one. We also give coset bounds for the involved codes.

On the distance of stabilizer quantum codes from J-affine variety codes

“…Occasionally, we enlarge our codes with Steane type procedures [48,32,19]. Our supporting classical codes have been recently introduced in a series of papers [17,18,19] and, generically, named J-affine variety codes; they are evaluating codes and codes of this type have been recently considered in the literature [23,2,21,26,22,20]. We devote Section 1 to recall them and to state two fundamental results, Propositions 1.2 and 1.3, which make it easy to decide about self-orthogonality for this class of codes, both with respect to Euclidean and Hermitian inner product.…”