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.

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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.

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From a rational convex polytope of dimension r ≥ 2 J.P. Hansen constructed an error correcting code of length n = (q −1) r over the finite field Fq. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the sense of Goppa. We compute the dimension of the code using cohomology. The minimum distance is estimated using intersection theory and mixed volumes, extending the methods of J.P. Hansen for plane polytopes. Finally we give a counterexample to Joyner's conjectures [10]. * Partially supported by MEC MTM2004-00958 (Spain). Address:

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [30,28]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. * The result in this paper is in part submitted for possible presentation in IEEE Information Theory Workshop (ITW 2014) [16]. † olav@math.aau.dk ‡ stefano@math.aau.dk § ryutaroh@rmatsumoto.org ¶ diego@math.aau.dk yuanluo@sjtu.edu.cn 1 Keywords: linear code, Feng-Rao bound, Hermitian code, one-point algebraic geometric code, relative dimension/length profile, relative generalized Hamming weight, secret sharing, wiretap channel of type II.and RGHW/RDLP was only recently reported. In particular, few classes of linear codes have been examined for their RGHW/RDLP. In this paper we study RGHW of general linear codes by the Feng-Rao approach [17], and explore its consequences for one-point algebraic geometry (AG) codes [43,22] and in particular the Hermitian codes [42,40,47].The present paper starts with a discussion of known results regarding linear ramp secret sharing schemes and it continues with demonstrating that the RGHWs can also be used to express the best case information leakage. The main result of the paper is a method to estimate RGHW of one-point algebraic geometric codes. This is done by carefully applying the Feng-Rao bounds [17] for primary [1] as well as dual [11,12,37,22,32,21] codes. From this we derive a relatively simple bound which uses information on the corresponding Weierstrass semigroup [24,8]. As shall be demonstrated for Hermitian codes the new bound is often sharp. Moreover, for the same codes the RGHW are often much larger than the corresponding generalized Hamming weights (GHW) [44] which means that studies of RGHW cannot be substituted by those of GHW.The paper is organized as follows. Section 2 describes the use of RGHW in connection with linear ramp secret sharing schemes, and in connection with communication over the wiretap channel of type II. In Section 3 we apply the theory to the special case of MDS codes. In Section 4 we show -at the level of general linear codes -how to employ the Feng-Rao bounds to estimate RGHW. This method is then applied to one-point algebraic geometric codes in Section 5. We investigate Hermitian codes in Section 6, and treat the corresponding ramp secret sharing schemes in Section 7.2 Ramp secret sharing schemes and wiretap channels of type II Ramp secret sharing schemes were introduced in [4,46]. Let F q be the finite field with q elements. A ramp secret sharing scheme with t-privacy and rreconstruction is an algorithm that, given an input s ∈ F ℓ q , outputs a vector x ∈ F n q , the vector of shares that we want to share among n players, such tha...

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [1] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [30] (See also [3]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition in [30] requires the use of differentials which was not needed in [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric 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.