On the similarities between generalized rank and Hamming weights and their applications to network coding

Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both cases have been established and proven in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the rank case yet, or in some cases have been proven after developing a different machinery. The aim of this paper is to further relate both weights and generalized weights, show that the results and proofs in both cases are usually essentially the same, and see the significance of these similarities in network coding. Some of the new results in the rank case also have new consequences in the Hamming case

Generalized weights: an anticode approach

In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then we characterize both generalized Hamming and rank weights of a code in terms of the intersection of the code with optimal anticodes in the respective metrics. Inspired by this description, we propose a new algebraic invariant, which we call "Delsarte generalized weights", for Delsarte rank-metric codes based on optimal anticodes of matrices. We show that our invariant refines the generalized rank weights for Gabidulin codes proposed by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of Delsarte generalized weights. In particular, we characterize Delsarte optimal codes and anticodes in terms of their generalized weights. We also present a duality theory for the new algebraic invariant, proving that the Delsarte generalized weights of a code completely determine the Delsarte generalized weights of the dual code. Our results extend the theory of generalized rank weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes of a theorem of Wei, proving that their generalized rank weights characterize the worst-case security drops of a Gabidulin rank-metric code

Higher Hamming weights for locally recoverable codes on algebraic curves

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015

End-to-End Error-Correcting Codes on Networks with Worst-Case Symbol Errors

The problem of coding for networks experiencing worst-case symbol errors is considered. We argue that this is a reasonable model for highly dynamic wireless network transmissions. We demonstrate that in this setup prior network error-correcting schemes can be arbitrarily far from achieving the optimal network throughput. A new transform metric for errors under the considered model is proposed. Using this metric, we replicate many of the classical results from coding theory. Specifically, we prove new Hamming-type, Plotkin-type, and Elias-Bassalygo-type upper bounds on the network capacity. A commensurate lower bound is shown based on Gilbert-Varshamov-type codes for error-correction. The GV codes used to attain the lower bound can be non-coherent, that is, they do not require prior knowledge of the network topology. We also propose a computationally-efficient concatenation scheme. The rate achieved by our concatenated codes is characterized by a Zyablov-type lower bound. We provide a generalized minimum-distance decoding algorithm which decodes up to half the minimum distance of the concatenated codes. The end-to-end nature of our design enables our codes to be overlaid on the classical distributed random linear network codes [1]. Furthermore, the potentially intensive computation at internal nodes for the link-by-link error-correction is un-necessary based on our design.Comment: Submitted for publication. arXiv admin note: substantial text overlap with arXiv:1108.239

Neural networks, error-correcting codes, and polynomials over the binary n-cube

Several ways of relating the concept of error-correcting codes to the concept of neural networks are presented. Performing maximum-likelihood decoding in a linear block error-correcting code is shown to be equivalent to finding a global maximum of the energy function of a certain neural network. Given a linear block code, a neural network can be constructed in such a way that every codeword corresponds to a local maximum. The connection between maximization of polynomials over the n-cube and error-correcting codes is also investigated; the results suggest that decoding techniques can be a useful tool for solving such maximization problems. The results are generalized to both nonbinary and nonlinear codes

Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding

By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C_1 of length n over a field extension and its subcode C_2. One is called the relative dimension/intersection profile (RDIP), and the other is called the relative generalized rank weight (RGRW). We clarify their basic properties and the relation between the RGRW and the minimum rank distance. As applications of the RDIP and the RGRW, the security performance and the error correction capability of secure network coding, guaranteed independently of the underlying network code, are analyzed and clarified. We propose a construction of secure network coding scheme, and analyze its security performance and error correction capability as an example of applications of the RDIP and the RGRW. Silva and Kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim C_1-1 links are wiretapped, which is guaranteed over any underlying network code. However, the explicit construction of such a scheme remained an open problem. Our new construction is just one instance of secure network coding that solves this open problem.Comment: IEEEtran.cls, 25 pages, no figure, accepted for publication in IEEE Transactions on Information Theor