8,600 research outputs found
A characterization of MDS codes that have an error correcting pair
Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were
found independently by R. K\"otter (1992), as a general algebraic method of
decoding linear codes. These pairs exist for several classes of codes. However
little or no study has been made for characterizing those codes. This article
is an attempt to fill the vacuum left by the literature concerning this
subject. Since every linear code is contained in an MDS code of the same
minimum distance over some finite field extension we have focused our study on
the class of MDS codes.
Our main result states that an MDS code of minimum distance has a
-ECP if and only if it is a generalized Reed-Solomon code. A second proof is
given using recent results Mirandola and Z\'emor (2015) on the Schur product of
codes
Subspace subcodes of Reed-Solomon codes
We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems
Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes
We give polynomial time attacks on the McEliece public key cryptosystem based
either on algebraic geometry (AG) codes or on small codimensional subcodes of
AG codes. These attacks consist in the blind reconstruction either of an Error
Correcting Pair (ECP), or an Error Correcting Array (ECA) from the single data
of an arbitrary generator matrix of a code. An ECP provides a decoding
algorithm that corrects up to errors, where denotes
the designed distance and denotes the genus of the corresponding curve,
while with an ECA the decoding algorithm corrects up to
errors. Roughly speaking, for a public code of length over ,
these attacks run in operations in for the
reconstruction of an ECP and operations for the reconstruction of an
ECA. A probabilistic shortcut allows to reduce the complexities respectively to
and . Compared to the
previous known attack due to Faure and Minder, our attack is efficient on codes
from curves of arbitrary genus. Furthermore, we investigate how far these
methods apply to subcodes of AG codes.Comment: A part of the material of this article has been published at the
conferences ISIT 2014 with title "A polynomial time attack against AG code
based PKC" and 4ICMCTA with title "Crypt. of PKC that use subcodes of AG
codes". This long version includes detailed proofs and new results: the
proceedings articles only considered the reconstruction of ECP while we
discuss here the reconstruction of EC
Shortened Array Codes of Large Girth
One approach to designing structured low-density parity-check (LDPC) codes
with large girth is to shorten codes with small girth in such a manner that the
deleted columns of the parity-check matrix contain all the variables involved
in short cycles. This approach is especially effective if the parity-check
matrix of a code is a matrix composed of blocks of circulant permutation
matrices, as is the case for the class of codes known as array codes. We show
how to shorten array codes by deleting certain columns of their parity-check
matrices so as to increase their girth. The shortening approach is based on the
observation that for array codes, and in fact for a slightly more general class
of LDPC codes, the cycles in the corresponding Tanner graph are governed by
certain homogeneous linear equations with integer coefficients. Consequently,
we can selectively eliminate cycles from an array code by only retaining those
columns from the parity-check matrix of the original code that are indexed by
integer sequences that do not contain solutions to the equations governing
those cycles. We provide Ramsey-theoretic estimates for the maximum number of
columns that can be retained from the original parity-check matrix with the
property that the sequence of their indices avoid solutions to various types of
cycle-governing equations. This translates to estimates of the rate penalty
incurred in shortening a code to eliminate cycles. Simulation results show that
for the codes considered, shortening them to increase the girth can lead to
significant gains in signal-to-noise ratio in the case of communication over an
additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information
Theory, Aug 200
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