1,887 research outputs found
On the cycle index and the weight enumerator II
In the previous paper, the second and third named author introduced the
concept of the complete cycle index and discussed a relation with the complete
weight enumerator in coding theory. In the present paper, we introduce the
concept of the complete joint cycle index and the average complete joint cycle
index, and discuss a relation with the complete joint weight enumerator and the
average complete joint weight enumerator respectively in coding theory.
Moreover, the notion of the average intersection numbers is given, and we
discuss a relation with the average intersection numbers in coding theory.Comment: 24 page
On THE AVERAGE JOINT CYCLE INDEX AND THE AVERAGE JOINT WEIGHT ENUMERATOR (Research on finite groups, algebraic combinatorics, and vertex algebras)
In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator respectively in coding theory
Geometric representations of linear codes
We say that a linear code C over a field F is triangular representable if
there exists a two dimensional simplicial complex such that C is a
punctured code of the kernel ker of the incidence matrix of
over F and there is a linear mapping between C and ker which is a
bijection and maps minimal codewords to minimal codewords. We show that the
linear codes over rationals and over GF(p), where p is a prime, are triangular
representable. In the case of finite fields, we show that this representation
determines the weight enumerator of C. We present one application of this
result to the partition function of the Potts model.
On the other hand, we show that there exist linear codes over any field
different from rationals and GF(p), p prime, that are not triangular
representable. We show that every construction of triangular representation
fails on a very weak condition that a linear code and its triangular
representation have to have the same dimension.Comment: 20 pages, 8 figures, v3 major change
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
Explicit Constructions of Quasi-Uniform Codes from Groups
We address the question of constructing explicitly quasi-uniform codes from
groups. We determine the size of the codebook, the alphabet and the minimum
distance as a function of the corresponding group, both for abelian and some
nonabelian groups. Potentials applications comprise the design of almost affine
codes and non-linear network codes
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
Iterative min-sum decoding of tail-biting codes
By invoking a form of the Perron-Frobenius theorem for the “min-sum” semi-ring, we obtain a union bound on the performance of iterative decoding of tail-biting codes. This bound shows that for the Gaussian channel, iterative decoding will be optimum, at least for high SNRs, if and only if the minimum “pseudo-distance” of the code is larger than the ordinary minimum distance
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View
These are the notes for a set of lectures delivered by the two authors at the
Les Houches Summer School on `Complex Systems' in July 2006. They provide an
introduction to the basic concepts in modern (probabilistic) coding theory,
highlighting connections with statistical mechanics. We also stress common
concepts with other disciplines dealing with similar problems that can be
generically referred to as `large graphical models'.
While most of the lectures are devoted to the classical channel coding
problem over simple memoryless channels, we present a discussion of more
complex channel models. We conclude with an overview of the main open
challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July
2006, 44 pages, 25 ps figure
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