21,425 research outputs found
Binary and Ternary Quasi-perfect Codes with Small Dimensions
The aim of this work is a systematic investigation of the possible parameters
of quasi-perfect (QP) binary and ternary linear codes of small dimensions and
preparing a complete classification of all such codes. First we give a list of
infinite families of QP codes which includes all binary, ternary and quaternary
codes known to is. We continue further with a list of sporadic examples of
binary and ternary QP codes. Later we present the results of our investigation
where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions
up to 13 are classified.Comment: 4 page
On Completely Regular Codes
This work is a survey on completely regular codes. Known properties,
relations with other combinatorial structures and constructions are stated. The
existence problem is also discussed and known results for some particular cases
are established. In particular, we present a few new results on completely
regular codes with covering radius 2 and on extended completely regular codes
On -extremal singly even self-dual codes
A relationship between -extremal singly even self-dual
codes and extremal doubly even self-dual
codes with covering radius meeting the Delsarte bound, is
established. As an example of the relationship, -extremal singly even
self-dual codes are constructed for the first time. In addition,
we show that there is no extremal doubly even self-dual code of length
with covering radius meeting the Delsarte bound for . Similarly, we
show that there is no extremal doubly even self-dual code of length
with covering radius meeting the Delsarte bound for .Comment: 15 pages, minor revisio
Completely regular codes by concatenating Hamming codes
We construct new families of completely regular codes by concatenation
methods. By combining parity check matrices of cyclic Hamming codes, we obtain
families of completely regular codes. In all cases, we compute the intersection
array of these codes. We also study when the extension of these codes gives
completely regular codes. Some of these new codes are completely transitive
New completely regular q-ary codes based on Kronecker products
For any integer and for any prime power q, the explicit
construction of a infinite family of completely regular (and completely
transitive) q-ary codes with d=3 and with covering radius is given. The
intersection array is also computed. Under the same conditions, the explicit
construction of an infinite family of q-ary uniformly packed codes (in the wide
sense) with covering radius , which are not completely regular, is also
given. In both constructions the Kronecker product is the basic tool that has
been used.Comment: Submitted to IT-IEEE. Theorem 1 in Section III was presented at the
2nd International Castle Meeting on Coding Theory and Applications (2ICMCTA),
Medina del Campo, Spain, September 2008.}
On Some Classes of Linear Codes and their Covering Radius
In this paper we define Simplex and MacDonald
Codes of type and and we give the covering radius of these
codes.Comment: arXiv admin note: text overlap with arXiv:1411.1822 by other author
On linear completely regular codes with covering radius . Construction and classification
Completely regular codes with covering radius must have minimum
distance . For , such codes are perfect and their parameters are
well known. In this paper, the cases and are studied and completely
characterized when the codes are linear. Moreover, it is proven that all these
codes are completely transitive.Comment: Submitted to IEEE, Trans. Inf. Theor
New Set of Codes for the Maximum-Likelihood Decoding Problem
The maximum-likelihood decoding problem is known to be NP-hard for general
linear and Reed-Solomon codes. In this paper, we introduce the notion of
A-covered codes, that is, codes that can be decoded through a polynomial time
algorithm A whose decoding bound is beyond the covering radius. For these
codes, we show that the maximum-likelihood decoding problem is reachable in
polynomial time in the code parameters. Focusing on bi- nary BCH codes, we were
able to find several examples of A-covered codes, including two codes for which
the maximum-likelihood decoding problem can be solved in quasi-quadratic time.Comment: in Yet Another Conference on Cryptography, Porquerolle : France
(2010
Simplex and MacDonald Codes over
In this paper, we introduce the homogeneous weight and homogeneous Gray map
over the ring for . We also
consider the construction of simplex and MacDonald codes of types and
over this ring
A complete classification of doubly even self-dual codes of length 40
A complete classification of binary doubly even self-dual codes of length 40
is given. As a consequence, a classification of binary extremal self-dual codes
of length 38 is also given.Comment: corrected typ
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