IA Scholar Query: Rank metric and Gabidulin codes in characteristic zero.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 26 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Extractors for Images of Varieties
https://scholar.archive.org/work/apfsslxulvatvfsurup6p7gqui
We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map f : 𝔽_q^r →𝔽_q^n to an element sampled uniformly at random from a k-dimensional variety V ⊆𝔽_q^r. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map f and the variety V, which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir-Gabizon-Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size.Zeyu Guo, Ben Lee Volk, Akhil Jalan, David Zuckermanwork_apfsslxulvatvfsurup6p7gquiSat, 26 Nov 2022 00:00:00 GMTConstructing MRD codes by switching
https://scholar.archive.org/work/5kk7kchoyvavzknwviql25blim
MRD codes are maximum codes in the rank-distance metric space on m-by-n matrices over the finite field of order q. They are diameter perfect and have the cardinality q^m(n-d+1) if m≥ n. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in m if the other parameters (n, q, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. Keywords: MRD codes, rank distance, bilinear forms graph, switching, diameter perfect codesMinjia Shi, Denis S. Krotov, Ferruh Özbudakwork_5kk7kchoyvavzknwviql25blimTue, 01 Nov 2022 00:00:00 GMTThe Euler characteristic, q-matroids, and a Möbius function
https://scholar.archive.org/work/32v5yk2pdbedjkkqlnx7rbb7pm
We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the Möbius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of q-matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a q-matroid complex and the lattice of q-cycles of the q-matroid. We use this formula to find the complete homology over ℤ of this shellable simplicial complex. We give a characterization of nonzero Euler characteristic for such order complexes. Finally, based on these results we remark why singular homology of a q-matroid equipped with order topology may not be effective to describe the q-cycles unlike the classical case of matroids.Trygve Johnsen, Rakhi Pratihar, Tovohery Hajatiana Randrianarisoawork_32v5yk2pdbedjkkqlnx7rbb7pmSat, 22 Oct 2022 00:00:00 GMTNew MRD codes from linear cutting blocking sets
https://scholar.archive.org/work/7wb2d2hmp5aq3d7ouy77z5n64q
AbstractMinimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated q-system. Using this result, we provide the first construction of a family of $$\mathbb F_{q^m}$$ F q m -linear MRD codes of length 2m that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.Daniele Bartoli, Giuseppe Marino, Alessandro Neriwork_7wb2d2hmp5aq3d7ouy77z5n64qSat, 16 Jul 2022 00:00:00 GMTLRPC codes with multiple syndromes: near ideal-size KEMs without ideals
https://scholar.archive.org/work/y4jwmsrbe5a4dn2tazn6c2abiy
We introduce a new rank-based key encapsulation mechanism (KEM) with public key and ciphertext sizes around 3.5 Kbytes each, for 128 bits of security, without using ideal structures. Such structures allow to compress objects, but give reductions to specific problems whose security is potentially weaker than for unstructured problems. To the best of our knowledge, our scheme improves in size all the existing unstructured post-quantum lattice or code-based algorithms such as FrodoKEM or Classic McEliece. Our technique, whose efficiency relies on properties of rank metric, is to build upon existing Low Rank Parity Check (LRPC) code-based KEMs and to send multiple syndromes in one ciphertext, allowing to reduce the parameters and still obtain an acceptable decoding failure rate. Our system relies on the hardness of the Rank Support Learning problem, a well-known variant of the Rank Syndrome Decoding problem. The gain on parameters is enough to significantly close the gap between ideal and non-ideal constructions. It enables to choose an error weight close to the rank Gilbert-Varshamov bound, which is a relatively harder zone for algebraic attacks. We also give a version of our KEM that keeps an ideal structure and permits to roughly divide the bandwidth by two compared to previous versions of LRPC KEMs submitted to the NIST with a Decoding Failure Rate (DFR) of 2^-128.Carlos Aguilar-Melchor, Nicolas Aragon, Victor Dyseryn, Philippe Gaborit, Gilles Zémorwork_y4jwmsrbe5a4dn2tazn6c2abiyThu, 23 Jun 2022 00:00:00 GMTGeneric Reed-Solomon codes achieve list-decoding capacity
https://scholar.archive.org/work/batoracxfrdkdl4qqhf46pgb4y
In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-ℓ MDS code, denoted by MDS(ℓ), has the property that any ℓ subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth defined a different notion of higher order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher order MDS codes are (nearly) equivalent. We also show that generic Reed-Solomon codes are MDS(ℓ) for all ℓ, relying crucially on the GM-MDS theorem which shows that generator matrices of generic Reed-Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed-Solomon codes achieve list decoding capacity. More concretely, we show that, with high probability, a random Reed-Solomon code of rate R over an exponentially large field is list decodable from radius 1-R-ϵ with list size at most 1-R-ϵ/ϵ, resolving a conjecture of Shangguan and Tamo.Joshua Brakensiek, Sivakanth Gopi, Visu Makamwork_batoracxfrdkdl4qqhf46pgb4yFri, 10 Jun 2022 00:00:00 GMTImproved Maximally Recoverable LRCs using Skew Polynomials
https://scholar.archive.org/work/76ia2qvbenfhhlhfdvevznjee4
An (n,r,h,a,q)-Local Reconstruction Code (LRC) is a linear code over 𝔽_q of length n, whose codeword symbols are partitioned into n/r local groups each of size r. Each local group satisfies 'a' local parity checks to recover from 'a' erasures in that local group and there are further h global parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it offers the best blend of locality and global erasure resilience – namely it can correct all erasure patterns whose recovery is information-theoretically feasible given the locality structure (these are precisely patterns with up to 'a' erasures in each local group and an additional h erasures anywhere in the codeword). Random constructions can easily show the existence of MR LRCs over very large fields, but a major algebraic challenge is to construct MR LRCs, or even show their existence, over smaller fields, as well as understand inherent lower bounds on their field size. We give an explicit construction of (n,r,h,a,q)-MR LRCs with field size q bounded by (O(max{r,n/r}))^min{h,r-a}. This improves upon known constructions in many relevant parameter ranges. Moreover, it matches the lower bound from Gopi et al. (2020) in an interesting range of parameters where r=Θ(√(n)), r-a=Θ(√(n)) and h is a fixed constant with h≤ a+2, achieving the optimal field size of Θ_h(n^h/2). Our construction is based on the theory of skew polynomials. We believe skew polynomials should have further applications in coding and complexity theory; as a small illustration we show how to capture algebraic results underlying list decoding folded Reed-Solomon and multiplicity codes in a unified way within this theory.Sivakanth Gopi, Venkatesan Guruswamiwork_76ia2qvbenfhhlhfdvevznjee4Wed, 18 May 2022 00:00:00 GMTOn subspace designs
https://scholar.archive.org/work/arnogzequrgezfvlpp4ocz5hfu
The aim of this paper is to investigate the theory of subspace designs, which have been originally introduced by Guruswami and Xing in 2013 to give the first construction of positive rate rank-metric codes list decodable beyond half the distance. However, sets of subspaces with special pattern of intersections with other subspaces have been already studied, such as spreads, generalised arcs and caps, and Cameron-Liebler sets. In this paper we provide bounds involving the dimension of the subspaces forming a subspace design and the parameters of the ambient space, showing constructions satisfying the equality in such bounds. Then we also introduce two dualities relations among them. Among subspace designs, those that we call s-designs are central in this paper, as they generalize the notion of s-scattered subspace to subspace design and for their special properties. Indeed, we prove that, for certain values of s, they correspond to the optimal subspace designs, that is those subspace designs that are associated with linear maximum sum-rank metric codes. Special attention has been paid for the case s=1 for which we provide several examples, yielding surprising to families of two-intersection sets with respect to hyperplanes (and hence two-weight linear codes). Moreover, s-designs can be used to construct explicit lossless dimension expanders (a linear-algebraic analogue of expander graphs), without any restriction on the order of the field. Another class of subspace designs we study is those of cutting designs, since they extend the notion of cutting blocking set recently introduced by Bonini and Borello. These designs turn out to be very interesting as they in one-to-one correspondence with minimal sum-rank metric codes. The latter codes have been introduced in this paper and they naturally extend the notions of minimal codes in both Hamming and rank metrics.Paolo Santonastaso, Ferdinando Zullowork_arnogzequrgezfvlpp4ocz5hfuWed, 27 Apr 2022 00:00:00 GMTSome Public-key Cryptosystems Over Finite Fields
https://scholar.archive.org/work/5ot7tjyppfhzxmhihdqaddoidq
In this paper, we present two public-key cry ptosystems over finite fields. First of them is based on polynomials. The presented system also considers a digital signature algorithm. Its security is based on the difficulty of finding discrete logarithms over GF(qd+1) with sufficiently large q and d. Is is also examined along with comparison with other polynomial based public-key systems. The other public-key cryptosystem is based on linear codes. McEliece studied the first code-based public-key cryptosystem. We are inspired by McEliece system in the construction of the new system. We examine its security using linear algebra and compare it with the other code-based cryptosystems. Our new cryptosystems are too reliable in terms of security.Selda Calkavurwork_5ot7tjyppfhzxmhihdqaddoidqTue, 26 Apr 2022 00:00:00 GMTq-Polymatroids and Their Relation to Rank-Metric Codes
https://scholar.archive.org/work/pajcm6u5lncdhflswxxypg2cti
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by F_q^m-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated q-polymatroid.Heide Gluesing-Luerssen, Benjamin Janywork_pajcm6u5lncdhflswxxypg2ctiThu, 10 Mar 2022 00:00:00 GMTTwisted Reed-Solomon Codes
https://scholar.archive.org/work/xqxwxc64jvfi5gm6qycxqip3im
In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed-Solomon codes. Whereas Reed-Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results.Peter Beelen, Sven Puchinger, Johan Rosenkildework_xqxwxc64jvfi5gm6qycxqip3imSun, 23 Jan 2022 00:00:00 GMTQuasi optimal anticodes: structure and invariants
https://scholar.archive.org/work/fjeaa4tpp5aopjzcycmquva7pu
It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.Elisa Gorla, Cristina Landolinawork_fjeaa4tpp5aopjzcycmquva7puWed, 19 Jan 2022 00:00:00 GMTConstructions of optimal rank-metric codes from automorphisms of rational function fields
https://scholar.archive.org/work/l6a7t2365vhrvd3fdfgjpa4wfq
<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>Rakhi Pratihar, Tovohery Hajatiana Randrianarisoawork_l6a7t2365vhrvd3fdfgjpa4wfqConstruction and bounds for subspace codes
https://scholar.archive.org/work/5fizllfmjbbqfcu6lsmygnp5gu
Subspace codes are the q-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes.Sascha Kurzwork_5fizllfmjbbqfcu6lsmygnp5guWed, 22 Dec 2021 00:00:00 GMT