We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.2010 Mathematics Subject Classification. 81P70; 94B65; 94B05.

Abstract. New stabilizer codes with parameters better than the ones available in the literature are provided in this work, in particular quantum codes with parameters [[127, 63, ≥ 12]]2 and [[63, 45, ≥ 6]]4 that are records. These codes are constructed with a new generalization of the Steane's enlargement procedure and by considering orthogonal subfield-subcodes -with respect to the Euclidean and Hermitian inner product-of a new family of linear codes, the J-affine variety codes.

Abstract. We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction. With this procedure we get codes with good parameters, some of them exceeding the quantum Gilbert-Varshamov bound given by Feng and Ma.

We study valuations centered in a regular local ring of dimension two. We define the notion of saturation with respect to such a valuation, extending the classical definitions. The invariants associated in a natural way to the valuation are related with the saturated ring and some geometric properties are deduced. IntroductionThe study of valuations in the context of Singularities was developed by Zariski and Abhyankar in the fifties in connection with the problem of resolution of singularities. Recently, Spivakovsky has revitalized valuation theory looking for progress in the same problem. In particular, [S2] gives a complete classification of valuations of F centered in a regular local ring R of dimension 2, F being the field of fractions of R. Spivakovsky has shown that the problem of classifying these valuations is essentially the same as that of classifying plane curve singularities in Spec R. In the classification process, invariants usually associated to curve singularities (see for instance [C1]) appear in a natural way: the semigroup of values, the set of values of maximal contact (minimal set of generators of such semigroups), the characteristic sequences, ... Finally, a purely algebraic treatment of the infinitely near singular points in connection with this classification problem can be found in a recent paper of Lipman ([L3]).

Abstract. We consider surfaces X defined by plane divisorial valuations ν of the quotient field of the local ring R at a closed point p of the projective plane P 2 over an arbitrary algebraically closed field k and centered at R. We prove that the regularity of the cone of curves of X is equivalent to the fact that ν is non positive on O P 2 (P 2 \ L), where L is a certain line containing p. Under these conditions, we characterize when the characteristic cone of X is closed and its Cox ring finitely generated. Equivalent conditions to the fact that ν is negative on O P 2 (P 2 \ L) \ k are also given.

We introduce the concept of δ‐sequence. A δ‐sequence Δ generates a well‐ordered semigroup S in ℤ2 or ℝ. We explain how to construct (and to compute parameters of) the dual code of any evaluation code associated with a weight function defined by Δ from the polynomial ring in two indeterminates to a semigroup S as above. We prove that this is a simple procedure that can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of δ‐sequences of the different existing types, and so this paper helps know and use a new, large set of codes.

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.2010 Mathematics Subject Classification. 81P70; 94B65; 94B05.

Two new constructions of linear code pairs C 2 ⊂ C 1 are given for which the codimension and the relative minimum distances M 1 (C 1 , C 2 ), M 1 (C ⊥ 2 , C ⊥ 1 ) are good. By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes [40]. They can also be used to ensure reliable communication over asymmetric quantum channels [54]. The new constructions result from carefully applying the Feng-Rao bounds [21,31] to a family of codes defined from multivariate polynomials and Cartesian product point sets.