31 research outputs found
Results on Binary Linear Codes With Minimum Distance 8 and 10
All codes with minimum distance 8 and codimension up to 14 and all codes with
minimum distance 10 and codimension up to 18 are classified. Nonexistence of
codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new
exact bounds for binary linear codes. Primarily two algorithms considering the
dual codes are used, namely extension of dual codes with a proper coordinate,
and a fast algorithm for finding a maximum clique in a graph, which is modified
to find a maximum set of vectors with the right dependency structure.Comment: Submitted to the IEEE Transactions on Information Theory, May 2010 To
be presented at the ACCT 201
What is Q-extension?
In this paper we present a developed software in the area of
Coding Theory. Using it, codes with given properties can be classified.
A part of this software can be used also for investigations (isomorphisms,
automorphism groups) of other discrete structures-combinatorial designs,
Hadamard matrices, bipartite graphs etc
Efficient Computing of some Vector Operations over GF(3) and GF(4)
The problem of efficient computing of the affine vector operations (addition of two vectors and multiplication of a vector by a scalar over GF (q)), and also the weight of a given vector, is important for many
problems in coding theory, cryptography, VLSI technology etc. In this paper
we propose a new way of representing vectors over GF (3) and GF (4) and
we describe an efficient performance of these affine operations. Computing
weights of binary vectors is also discussed
Algorithms for Computing the Linearity and Degree of Vectorial Boolean Functions
In this article, we study two representations of a Boolean function
which are very important in the context of cryptography. We describe
Möbius and Walsh Transforms for Boolean functions in details and present
effective algorithms for their implementation. We combine these algorithms
with the Gray code to compute the linearity, nonlinearity and algebraic degree
of a vectorial Boolean function. Such a detailed consideration will be
very helpful for students studying the design of block ciphers, including PhD
students in the beginning of their research.
ACM Computing Classification System (1998): F.2.1, F.2.2
Representing Equivalence Problems for Combinatorial Objects
Methods for representing equivalence problems of various combinatorial objects
as graphs or binary matrices are considered. Such representations can be used
for isomorphism testing in classification or generation algorithms.
Often it is easier to consider a graph or a binary matrix isomorphism problem
than to implement heavy algorithms depending especially on particular combinatorial
objects. Moreover, there already exist well tested algorithms for the graph isomorphism
problem (nauty) and the binary matrix isomorphism problem as well (Q-Extension).
ACM Computing Classification System (1998): F.2.1, G.4
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
Optimal binary linear codes of dimension at most seven
AbstractWe classify optimal [n,k,d] binary linear codes of dimension ⩽7, with one exception, where by optimal we mean that no [n−1,k,d],[n+1,k+1,d], or [n+1,k,d+1] code exists. In particular, we present (new) classification results for codes with parameters [40,7,18], [43,7,20], [59,7,28], [75,7,36], [79,7,38], [82,7,40], [87,7,42], and [90,7,44]. These classifications are accomplished with the aid of the first author's computer program Extension for extending from residual codes, and the second author's program Split
Пресмятане на минималното разстояние на линеен код
Some aspects of the algorithms for calculating the minimum distance of linear codes over finite fields are presented.The research of Iliya Bouyukliev was supported, in part, by a Bulgarian NSF contract KP-060-Russia/33/17.12.202