Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of Fqn under the act...
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Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of Fqn under the action of some subgroup of the finite general linear group GLn(q). The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of GLn(q), we propose a new construction of an abelian non-cyclic orbit codes of size qk with k <= n/2. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle functi...
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We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function f : L-m -> F-q, where L subset of F-q, a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed-Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed-Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed-Solomon codes of [Ben-Sasson et al., ICALP 2018] named "FRI protocol". Compared to the FRI protocol, our IOPP for tensor products of Reed-Solomon codes achieves the same efficiency parameters. As for Reed-Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed-Muller codes compares well, all things equal.
The covering radius of a q-ary block code C of length n is defined as the smallest integer R = R(C) such that all vectors in F-q(n) are within Hamming distance R of some codeword of C. By En, k, d]R code, we mean an E...
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The covering radius of a q-ary block code C of length n is defined as the smallest integer R = R(C) such that all vectors in F-q(n) are within Hamming distance R of some codeword of C. By En, k, d]R code, we mean an En, k, d] code having covering radius R. The covering radius of a code is one of the fundamental parameters of a code and gives its suitability for data compression, list decoding radius, and has many other applications. The upper bound of Janwa (1986) relates all the fundamental parameters as R(C) <= t'(C) := n Sigma(k)(i=1) left perpendiculard/2(i) right perdpendicular. Which can be expressed as n - g(q) + d [d/q(k)]. If n(q)(k, d) denotes the minimum length of any code of dimension k and distance over F-q it was conjectured by Janwa that under certain conditions g(q) (k, d) (the Griesmer length) can be replaced by nq (k, d). Janwa (1989) and Janwa and Mattson (1999), proved three of the four cases and conjectured that the final case is true. In this article, we give a resolution of this conjecture. These bounds have helped us in determing the exact covering radius of codes from Hermitian curves in most cases, and yielding close bounds in the rest of them.
Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of F-q(n) under the action of...
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Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of F-q(n) under the action of some subgroup of the finite general linear group GL(n)(q). The aim of this paper is to present constructions of large non-Abelian orbit codes having the maximum possible distance. The properties of imprimitive wreath products and wreathed tensor products of groups are employed to select certain types of subspaces and their stabilizers, thereby providing a systematic way of constructing orbit codes with optimum parameters. We also present explicit examples of such constructions which improve the parameters of the construction already obtained in Climent et al. (Cryptogr Commun 11:839-852, 2019).
In this work we propose a family of Fq-linear low-rank parity check (LRPC) codes based on a bilinear product over F-q(m) defined by a generic 3-tensor over Fq. A particular choice of this tensor corresponds to the cla...
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ISBN:
(纸本)9798350301496
In this work we propose a family of Fq-linear low-rank parity check (LRPC) codes based on a bilinear product over F-q(m) defined by a generic 3-tensor over Fq. A particular choice of this tensor corresponds to the classical F-qm-linear LRPC codes;and other tensors yield Fq-linear codes, which, with some caveats, can be efficiently decoded with the same idea of decoding LRPC codes. The proposed codes contribute to the diversity of rank metric codes for cryptographic applications, particularly for the cases where attacks utilize F-qm-linearity to reduce decoding complexity.
Array codes are the preferred codes for distributed storage, such that different rows in an array are stored at different nodes. Layered codes use a sparse format for stored arrays with a single parity check per colum...
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Array codes are the preferred codes for distributed storage, such that different rows in an array are stored at different nodes. Layered codes use a sparse format for stored arrays with a single parity check per column and no other parity checks. Remarkably, the simple structure of layered codes is optimal when data is collected from all but one node. Codes that collect data from fewer nodes include improved layered codes, determinant codes, cascade codes and moulin codes. As our main result we show that the concatenation of layered codes with suitable outer codes achieves the performance of cascade and moulin codes which is conjectured to be optimal for general regenerating codes. The codes that we use as outer codes are in a new class of codes that we call Johnson graph codes. The codes have properties similar to those of Reed-Muller codes. In both cases the topological structure of the set of coordinates can be used to identify information sets and codewords of small weight.
LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanne...
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LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes. Copyright (C) 2022 The Authors.
Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic ap...
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Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf theory 65(12):7718-7735, 2019), we define and study LRPC codes over Galois rings-a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.
We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motiv...
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We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.
We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A q...
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We discuss the connection between quantum error-correcting codes (QECCS) and algebraic coding theory. We start with an introduction to the relevant concepts of quantum mechanics, including the general error model. A quantum error-correcting code is a subspace of a complex Hilbert space, and its error-correcting properties are characterized by the Knill-Laflamme conditions. Using the stabilizer formalism, we illustrate how QECCs for can be constructed using techniques from algebraic coding theory. We also sketch how the information obtained via a quantum measurement can be interpreted as syndrome of the related classical code. Additionally, we present secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs. This includes the puncture code by Rains and construction X for quantum codes.
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