270 research outputs found
Structural Properties of Twisted Reed-Solomon Codes with Applications to Cryptography
We present a generalisation of Twisted Reed-Solomon codes containing a new
large class of MDS codes. We prove that the code class contains a large
subfamily that is closed under duality. Furthermore, we study the Schur squares
of the new codes and show that their dimension is often large. Using these
structural properties, we single out a subfamily of the new codes which could
be considered for code-based cryptography: These codes resist some existing
structural attacks for Reed-Solomon-like codes, i.e. methods for retrieving the
code parameters from an obfuscated generator matrix.Comment: 5 pages, accepted at: IEEE International Symposium on Information
Theory 201
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions
Regenerating codes are a class of recently developed codes for distributed
storage that, like Reed-Solomon codes, permit data recovery from any arbitrary
k of n nodes. However regenerating codes possess in addition, the ability to
repair a failed node by connecting to any arbitrary d nodes and downloading an
amount of data that is typically far less than the size of the data file. This
amount of download is termed the repair bandwidth. Minimum storage regenerating
(MSR) codes are a subclass of regenerating codes that require the least amount
of network storage; every such code is a maximum distance separable (MDS) code.
Further, when a replacement node stores data identical to that in the failed
node, the repair is termed as exact.
The four principal results of the paper are (a) the explicit construction of
a class of MDS codes for d = n-1 >= 2k-1 termed the MISER code, that achieves
the cut-set bound on the repair bandwidth for the exact-repair of systematic
nodes, (b) proof of the necessity of interference alignment in exact-repair MSR
codes, (c) a proof showing the impossibility of constructing linear,
exact-repair MSR codes for d < 2k-3 in the absence of symbol extension, and (d)
the construction, also explicit, of MSR codes for d = k+1. Interference
alignment (IA) is a theme that runs throughout the paper: the MISER code is
built on the principles of IA and IA is also a crucial component to the
non-existence proof for d < 2k-3. To the best of our knowledge, the
constructions presented in this paper are the first, explicit constructions of
regenerating codes that achieve the cut-set bound.Comment: 38 pages, 12 figures, submitted to the IEEE Transactions on
Information Theory;v3 - The title has been modified to better reflect the
contributions of the submission. The paper is extensively revised with
several carefully constructed figures and example
Systematic maximum sum rank codes
In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.publishe
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