32 research outputs found
On Algebraic Decoding of -ary Reed-Muller and Product-Reed-Solomon Codes
We consider a list decoding algorithm recently proposed by Pellikaan-Wu
\cite{PW2005} for -ary Reed-Muller codes of
length when . A simple and easily accessible
correctness proof is given which shows that this algorithm achieves a relative
error-correction radius of . This is
an improvement over the proof using one-point Algebraic-Geometric codes given
in \cite{PW2005}. The described algorithm can be adapted to decode
Product-Reed-Solomon codes.
We then propose a new low complexity recursive algebraic decoding algorithm
for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a
relative error correction radius of . This technique is then proved to outperform the Pellikaan-Wu
method in both complexity and error correction radius over a wide range of code
rates.Comment: 5 pages, 5 figures, to be presented at 2007 IEEE International
Symposium on Information Theory, Nice, France (ISIT 2007
List decoding of a class of affine variety codes
Consider a polynomial in variables and a finite point ensemble . When given the leading monomial of with respect to
a lexicographic ordering we derive improved information on the possible number
of zeros of of multiplicity at least from . We then use this
information to design a list decoding algorithm for a large class of affine
variety codes.Comment: 11 pages, 5 table
The attackers power boundaries for traceability of algebraic geometric codes on special curves
Под схемами широковещательного шифрования понимают такие протоколы распространения легально тиражируемой цифровой продукции, которые способны предотвратить несанкционированный доступ к распространяемым данным. Эти схемы широко используются как для распределённого хранения данных, так и для защиты данных при передаче по каналам связи, и исследование таких схем представляется актуальной задачей. Для предотвращения коалиционных атак в схемах широковещательного шифрования используются классы помехоустойчивых кодов со специальными свойствами, в частности c-FP- и c-TA-свойствами. Рассматривается задача оценки нижней и верхней границ мощности коалиции злоумышленников, в пределах которых алгеброгеометрические коды обладают этими свойствами. Ранее были получены границы для одноточечных алгеброгеометрических кодов на кривых общего вида. В работе эти границы уточняются для одноточечных кодов на кривых специального вида; в частности, для кодов на кривых, на которых имеется достаточно много классов эквивалентности после факторизации множества точек кривой по отношению равенства соответствующих координат
Some remarks on multiplicity codes
Multiplicity codes are algebraic error-correcting codes generalizing
classical polynomial evaluation codes, and are based on evaluating polynomials
and their derivatives. This small augmentation confers upon them better local
decoding, list-decoding and local list-decoding algorithms than their classical
counterparts. We survey what is known about these codes, present some
variations and improvements, and finally list some interesting open problems.Comment: 21 pages in Discrete Geometry and Algebraic Combinatorics, AMS
Contemporary Mathematics Series, 201
Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex
Reed-Muller codes are among the most important classes of locally correctable
codes. Currently local decoding of Reed-Muller codes is based on decoding on
lines or quadratic curves to recover one single coordinate. To recover multiple
coordinates simultaneously, the naive way is to repeat the local decoding for
recovery of a single coordinate. This decoding algorithm might be more
expensive, i.e., require higher query complexity. In this paper, we focus on
Reed-Muller codes with usual parameter regime, namely, the total degree of
evaluation polynomials is , where is the code alphabet size
(in fact, can be as big as in our setting). By introducing a novel
variation of codex, i.e., interleaved codex (the concept of codex has been used
for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover
arbitrarily large number of coordinates of a Reed-Muller code
simultaneously at the cost of querying coordinates. It turns out that
our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that
accessing locations is in fact cheaper than repeating the procedure for
accessing a single location for times. Our estimation of success error
probability is based on error probability bound for -wise linearly
independent variables given in \cite{BR94}