2,447 research outputs found
Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes
In this paper, we establish a lemma in algebraic coding theory that
frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes,
algebraic geometry codes, and affine variety codes. Our lemma corresponds to
the non-systematic encoding of affine variety codes, and can be stated by
giving a canonical linear map as the composition of an extension through linear
feedback shift registers from a Grobner basis and a generalized inverse
discrete Fourier transform. We clarify that our lemma yields the error-value
estimation in the fast erasure-and-error decoding of a class of dual affine
variety codes. Moreover, we show that systematic encoding corresponds to a
special case of erasure-only decoding. The lemma enables us to reduce the
computational complexity of error-evaluation from O(n^3) using Gaussian
elimination to O(qn^2) with some mild conditions on n and q, where n is the
code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE
Transactions on Information Theory on Jan. 8, 201
Decoding Reed-Muller codes over product sets
We give a polynomial time algorithm to decode multivariate polynomial codes
of degree up to half their minimum distance, when the evaluation points are
an arbitrary product set , for every . Previously known
algorithms can achieve this only if the set has some very special algebraic
structure, or if the degree is significantly smaller than . We also
give a near-linear time randomized algorithm, which is based on tools from
list-decoding, to decode these codes from nearly half their minimum distance,
provided .
Our result gives an -dimensional generalization of the well known decoding
algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic
version of the Schwartz-Zippel lemma.Comment: 25 pages, 0 figure
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
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