3,576 research outputs found
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 use of blocking sets in Galois geometries and in related research areas
Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
LDPC codes associated with linear representations of geometries
We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K
Large weight code words in projective space codes
AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points
Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries
In this paper we construct infinite families of non-linear maximum rank
distance codes by using the setting of bilinear forms of a finite vector space.
We also give a geometric description of such codes by using the cyclic model
for the field reduction of finite geometries and we show that these families
contain the non-linear maximum rank distance codes recently provided by
Cossidente, Marino and Pavese.Comment: submitted; 22 page
Abstract algebra, projective geometry and time encoding of quantum information
Algebraic geometrical concepts are playing an increasing role in quantum
applications such as coding, cryptography, tomography and computing. We point
out here the prominent role played by Galois fields viewed as cyclotomic
extensions of the integers modulo a prime characteristic . They can be used
to generate efficient cyclic encoding, for transmitting secrete quantum keys,
for quantum state recovery and for error correction in quantum computing.
Finite projective planes and their generalization are the geometric counterpart
to cyclotomic concepts, their coordinatization involves Galois fields, and they
have been used repetitively for enciphering and coding. Finally the characters
over Galois fields are fundamental for generating complete sets of mutually
unbiased bases, a generic concept of quantum information processing and quantum
entanglement. Gauss sums over Galois fields ensure minimum uncertainty under
such protocols. Some Galois rings which are cyclotomic extensions of the
integers modulo 4 are also becoming fashionable for their role in time encoding
and mutual unbiasedness.Comment: To appear in R. Buccheri, A.C. Elitzur and M. Saniga (eds.),
"Endophysics, Time, Quantum and the Subjective," World Scientific, Singapore.
16 page
Non-linear MRD codes from cones over exterior sets
By using the notion of -embedding of a (canonical) subgeometry
and of exterior set with respect to the -secant variety
of a subset , , in
the finite projective space , , in this article
we construct a class of non-linear -MRD codes for any . A code of this class, where and is a generator of
, arises from a cone of
with vertex an -dimensional subspace over a
maximum exterior set with respect to . We
prove that the codes introduced in [Cossidente, A., Marino, G., Pavese, F.:
Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597--609
(2016); Durante, N., Siciliano, A.: Non-linear maximum rank distance codes in
the cyclic model for the field reduction of finite geometries. Electron. J.
Comb. (2017); Donati, G., Durante, N.: A generalization of the normal rational
curve in and its associated non-linear MRD codes. Des.
Codes Cryptogr. 86, 1175--1184 (2018)] are appropriate punctured ones of
and solve completely the inequivalence issue for this
class showing that is neither equivalent nor adjointly
equivalent to the non-linear MRD code , , obtained in [Otal, K., \"Ozbudak, F.: Some new
non-additive maximum rank distance codes. Finite Fields and Their Applications
50, 293--303 (2018).]
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