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The Discovery of Simple 7Designs with Automorphism Group ...
, 1995
"... A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly ..."
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Cited by 10 (7 self)
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A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.
Isometry Classes of Indecomposable Linear Codes
 Proc. Int. Symp., AAECC11, Paris 1995, volume 948 of Lecture
, 1995
"... In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian. ..."
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Cited by 9 (2 self)
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In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian.
On the Voronoi Neighbor Ratio for Binary Linear Block Codes
 IEEE Trans. Inform. Theory
, 1998
"... Softdecision decoding of block codes is regarded as the geometrical problem of identifying the Voronoi region within which a given input vector lies. A measure, called the neighbor ratio, is proposed to characterize how many facets a Voronoi region has. Theory and algorithms are presented to determ ..."
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Cited by 6 (1 self)
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Softdecision decoding of block codes is regarded as the geometrical problem of identifying the Voronoi region within which a given input vector lies. A measure, called the neighbor ratio, is proposed to characterize how many facets a Voronoi region has. Theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes. An asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1/2 or not. For rates below this threshold, all pairs of codewords tend to share a Voronoi facet; for higher rates, a relatively small fraction of them do. Index TermsBinary linear block codes, Gaussian channel, Voronoi regions, neighbor ratio, asymptotic properties, softdecision decoding. I. INTRODUCTION A channel decoder is, in its common form, a device that receives a sequence of values from the demodulator and outputs another sequence, selected from a predefined set...
Classifying Subspaces of Hamming Spaces
"... A linear code in F n q with dimension k and minimum distance at least d is called an [n; k; d] q code. We here consider the problem of classifying all [n; k; d] q codes given n, k, d, and q. In other words, given the Hamming space F n q and a dimension k, we classify all k dimensional subsp ..."
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Cited by 6 (0 self)
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A linear code in F n q with dimension k and minimum distance at least d is called an [n; k; d] q code. We here consider the problem of classifying all [n; k; d] q codes given n, k, d, and q. In other words, given the Hamming space F n q and a dimension k, we classify all k dimensional subspaces of the Hamming space with minimum distance at least d. Our classication is an iterative procedure where equivalent codes are identied by mapping the code equivalence problem into the graph isomorphism problem, which is solved using the program nauty. For d = 3, the classication is explicitly carried out for binary codes of length n 14, ternary codes of length n 11, and quaternary codes of length n 10.
Enumeration of linear codes by applying methods from algebraic combinatorics
 Grazer Math. Ber
, 1996
"... It is demonstrated how classes of linear (n, k)codes can be enumerated using cycle index polynomials and other methods from algebraic combinatorics. Some results of joined work [9] with Prof. Kerber from the University of Bayreuth on the enumeration of linear codes over GF (q) are presented. Furthe ..."
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Cited by 5 (0 self)
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It is demonstrated how classes of linear (n, k)codes can be enumerated using cycle index polynomials and other methods from algebraic combinatorics. Some results of joined work [9] with Prof. Kerber from the University of Bayreuth on the enumeration of linear codes over GF (q) are presented. Furthermore I will give an introduction to enumeration under finite group actions. At first let me draw your attention to the enumeration of linear codes. Let p be a prime and let q be a power of p then GF (q) denotes the finite field of q elements. A linear (n, k)code over the Galois field GF (q) is a kdimensional subspace of the vector space GF (q) n. As usual codewords will be written as rows x = (x1,..., xn). A k × nmatrix Γ over GF (q) is called a generator matrix of the linear (n, k)code C, if and only if the rows of Γ form a basis of C, so that C = {x · Γ  x ∈ GF (q) k}. The Hamming distance d(x, y): = {i ∈ n⎪xi � = yi}  is a metric on GF (q) n. (The set of integers from 1 to n will be indicated as n.) The minimal distance d(C) of a code C is given by d(C): = min
Constructions of Mixed Covering Codes
, 1991
"... : In this work construction methods for so called mixed covering codes are developed. There has been considerable recent growth in the interest in covering codes. They have the property that all words in the space are within a given Hamming distance, called the covering radius, from some codeword. T ..."
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: In this work construction methods for so called mixed covering codes are developed. There has been considerable recent growth in the interest in covering codes. They have the property that all words in the space are within a given Hamming distance, called the covering radius, from some codeword. Traditionally, mainly spaces where all coordinates have the same arity (usually 2 or 3, that is, they are binary or ternary) have been discussed. In this work we consider mixed spaces F n 1 q 1 F n 2 q 2 \Delta \Delta \Delta F nm q m , where the coordinates can be of varying arities. The approach is very general, no restrictions are set upon m and the arities q i . The construction methods consist of generalizations of known constructions for covering codes and some completely new constructions. They are divided into three classes; direct constructions, constructions of new codes from old, and the matrix method. Through these constructions upper bounds for the minimal number of codewords...
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
Construction of quasicyclic codes
"... The class of QuasiCyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p QuasiCyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new code ..."
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The class of QuasiCyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p QuasiCyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some sth Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting
Some ErrorCorrecting Codes and Their Applications
"... Introduction In this chapter we describe three types of errorcorrecting linear codes that have been used in major applications, viz. photographs from spacecraft (first order ReedMuller codes), compact discs (ReedSolomon codes), and computer memories (extended binary Hamming codes). Errorcorrect ..."
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Introduction In this chapter we describe three types of errorcorrecting linear codes that have been used in major applications, viz. photographs from spacecraft (first order ReedMuller codes), compact discs (ReedSolomon codes), and computer memories (extended binary Hamming codes). Errorcorrecting codes were first developed in the 1940s following a theorem of Claude Shannon [14] that showed that almost errorfree communication could be obtained over a noisy channel. The message to be communicated is first "encoded", i.e. turned into a codeword, by adding "redundancy". The codeword is then sent through the channel and the received message is "decoded" by the receiver into a message resembling, as closely as possible, the original message. The degree of resemblance will depend on how good the code is in relation to the channel. Such codes have been used to great e#ect in some important applications, and we will describe here the codes that are used in three of these applications, s
Construction Methods for Covering Codes
, 1993
"... A covering code in a Hamming space is a set of codewords with the property that any word in the space is within a specified Hamming distance, the covering radius, from at least one codeword. In this thesis, efficient construction methods for such codes are considered. The constructions work, with so ..."
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A covering code in a Hamming space is a set of codewords with the property that any word in the space is within a specified Hamming distance, the covering radius, from at least one codeword. In this thesis, efficient construction methods for such codes are considered. The constructions work, with some exceptions, for codes over alphabets consisting of any number of symbols. Codes over mixed alphabets are also discussed. Most of the methods are developed in order to determine values of K q (n; R), the minimumnumber of codewords in a qary code of length n and covering radius R. Codes obtained by the constructions prove upper bounds on this function. In many of the constructions simulated annealing, a probabilistic optimization method, has turned out to perform very well. Simulated annealing cannot be used to prove optimality of codes found; in that case, the problem is viewed and solved as a set covering problem. For larger codes, a direct approach is not generally feasible; it is sho...