751 research outputs found
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Self-dual codes, subcode structures, and applications.
The classification of self-dual codes has been an extremely active area in coding theory since 1972 [33]. A particularly interesting class of self-dual codes is those of Type II which have high minimum distance (called extremal or near-extremal). It is notable that this class of codes contains famous unique codes: the extended Hamming [8,4,4] code, the extended Golay [24,12,8] code, and the extended quadratic residue [48,24,12] code. We examine the subcode structures of Type II codes for lengths up to 24, extremal Type II codes of length 32, and give partial results on the extended quadratic residue [48,24,12] code. We also develop a generalization of self-dual codes to Network Coding Theory and give some results on existence of self-dual network codes with largest minimum distance for lengths up to 10. Complementary Information Set (CIS for short) codes, a class of classical codes recently developed in [7], have important applications to Cryptography. CIS codes contain self-dual codes as a subclass. We give a new classification result for CIS codes of length 14 and a partial result for length 16
On the Spectrum of Wenger Graphs
Let , where is a prime and is an integer. For ,
let and be two copies of the -dimensional vector spaces over the
finite field . Consider the bipartite graph with partite
sets and defined as follows: a point is adjacent to a line if and only if the
following equalities hold: for . We call the graphs Wenger graphs. In this paper, we determine all
distinct eigenvalues of the adjacency matrix of and their
multiplicities. We also survey results on Wenger graphs.Comment: 9 pages; accepted for publication to J. Combin. Theory, Series
On Multivariate Cryptosystems Based on Computable Maps with Invertible Decomposition
Let K be a commutative ring and K^n be a space over K of dimension n. Weintroduce the concept of a family of multivariate maps f(n) of K^n into itself with invertible decomposition.If f(n) is computable in polynomial time then it can be used as the public rule and theinvertible decomposition provides a private key in f(n) based public key infrastructure. Requirementsof polynomial ity of degree and density for f(n) allow to estimate the complexity of encryption procedurefor a public user. The concepts of a stable family and a family of increasing order are motivatedby the studies of discrete logarithm problem in the Cremona group. The statement on the existenceof families of multivariate maps of polynomial degree and polynomial density of increasing order withthe invertible decomposition is proved. The proof is supported by explicite construction which canbe used as a new cryptosystem. The presented multivariate encryption maps are induced by specialwalks in the algebraically dened extremal graphs A(n;K) and D(n;K) of increasing girth
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
The implementation of cubic public keys based on a new family of algebraic graphs
Families of edge transitive algebraic graphs defined over finite commutative rings were used for the development of stream ciphers, public key cryptosystems and key exchange protocols. We present the results of the first implementation of a public key algorithm based on the family of algebraic graphs, which are not edge transitive. The absence of an edge transitive group of symmetries means that the algorithm can not be described in group theoretical terms. We hope that it licates cryptanalysis of the algorithm. We discuss the connections between the security of algorithms and the discrete logarithm problem.The plainspace of the algorithm is Kn, where K is the chosen commutative ring. The graph theoretical encryption corresponds to walk on the bipartite graph with the partition sets which are isomorphic to Kn. We conjugate the chosen graph based encryption map, which is a composition of several elementary cubical polynomial automorphisms of a free module Kn with special invertible affine transformation of Kn. Finally we compute symbolically the corresponding cubic public map g of Kn onto Kn. We evaluate time for the generation of g, and the number of monomial expression in the list of corresponding public rules
Additive Asymmetric Quantum Codes
We present a general construction of asymmetric quantum codes based on
additive codes under the trace Hermitian inner product. Various families of
additive codes over \F_{4} are used in the construction of many asymmetric
quantum codes over \F_{4}.Comment: Accepted for publication March 2, 2011, IEEE Transactions on
Information Theory, to appea
- …