The Correspondence Between Projective Codes and 2-weight Codes

We show how to get a 1-1 correspondence between projective linear codes and 2-weight linear codes. A generalization of the construction gives rise to several new ternary linear codes of dimension six.     

Ternary Covering Codes Derived from BCH Codes

It is shown how ternary BCH codes can be lengthened to get linear codes with covering radius 2. The family obtained has the ternary Golay code as its first code, contains codes with record-breaking parameters, and has a good asymptotic behavior. The ternary Golay code is further used to obtain short proofs for the best known upper bounds for the football pool problem for 11 and 12 matches.     

Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices

In this paper, a construction of ternary self-dual codes based on negacirculant matrices is given. As an application, we construct new extremal ternary self-dual codes of lengths 32, 40, 44, 52 and 56. Our approach regenerates all the known extremal self-dual codes of lengths 36, 48, 52 and 64. New extremal ternary quasi-twisted self-dual codes are also constructed. Supported by an NSERC discovery grant and a RTI grant. Supported by an NSERC discovery grant and a RTI grant. A summer student Chinook Scholarship is greatly appreciated.     

Orthogonal designs,self‐dual codes,and the Leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self‐dual codes. Orthogonal designs are shown to be useful in construction of self‐dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self‐dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.     

Improved Bounds for Ternary Linear Codes of Dimension 8 Using Tabu Search

In this paper, new codes of dimension 8 are presented which give improved bounds on the maximum possible minimum distance of ternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. Twenty three codes are given which improve or establish the bounds for ternary codes. In addition, a table of upper and lower bounds for d ₃(n, 8) is presented for n 200.     

On the Covering Radius of Ternary Extremal Self-Dual Codes

In this paper, we investigate the covering radius of ternary extremal self-dual codes. The covering radii of all ternary extremal self-dual codes of lengths up to 20 were previously known. The complete coset weight distributions of the two inequivalent extremal self-dual codes of length 24 are determined. As a consequence, it is shown that every extremal ternary self-dual code of length up to 24 has covering radius which meets the Delsarte bound. The first example of a ternary extremal self-dual code with covering radius which does not meet the Delsarte bound is also found. It is worth mentioning that the found code is of length 32.     

A New Table of Binary/Ternary Mixed Covering Codes

A table of upper bounds for K3,2(n1,n2;R), the minimum number of codewords in a covering code with n1 ternary coordinates, n2 binary coordinates, and covering radius R, in the range n = n1 + n2 13, R 3, is presented. Explicit constructions of codes are given to prove the new bounds and verify old bounds. These binary/ternary covering codes can be used as systems for the football pool game. The results include a new binary code with covering radius 1 proving K2(13,1) 736, and the following upper bound for the football pool problem for 9 matches: K3(9,1) 1356.     

Trellis Structure and Higher Weights of Extremal Self-Dual Codes

A method for demonstrating and enumerating uniformly efficient (permutation-optimal) trellis decoders for self-dual codes of high minimum distance is developed. Such decoders and corresponding permutations are known for relatively few codes.The task of finding such permutations is shown to be substantially simplifiable in the case of self-dual codes in general, and for self-dual codes of sufficiently high minimum distance it is shown that it is frequently possible to deduce the existence of these permutations directly from the parameters of the code.A new and tighter link between generalized Hamming weights and trellis representations is demonstrated: for some self-dual codes, knowledge of one of the generalized Hamming weights is sufficient to determine the entire optimal state complexity profile.These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32,16,8] binary self-dual codes and for several other codes. The numbers of uniformly efficient permutations for several codes, including the [24,12,8] Golay code and both [24,12,9] ternary self-dual codes, are found.     

On Perfect Ternary Constant Weight Codes

We consider the space of ternary words of length n and fixed weight w with the usual Hamming distance. A sequence of perfect single error correcting codes in this space is constructed. We prove the nonexistence of such codes with other parameters than those of the sequence.     

The Nonexistence of Ternary [50,5,32] Codes

It is unknown (cf. Hill and Newton [8] or Hamada [3]) whether or not there exists a ternary [50,5,32] code meeting the Griesmer bound. The purpose of this paper is to prove the nonexistence of ternary [50,5,32] codes. Since there exists a ternary [51,5,32] code, this implies that n3(5,32) = 51, where n3(k,d) denotes the smallest value of n for which there exists a ternary [n,k,d] code.     

A Construction of Optimal Constant Composition Codes

In this paper, a construction of optimal constant composition codes is developed, and used to derive some series of new optimal constant composition codes meeting the upper bound given by [13].     

The Covering Radius of Ternary Cyclic Codes with Length up to 25

The covering radius of all ternary cyclic codes of length up to 25 is given. Some of the results were obtained by computer and for others mathematical reasonings were applied. The minimal distances of all codes were recalculated.     

A Class of Perfect Ternary Constant-Weight Codes

We construct a class of perfect ternary constant-weight codes of length 2 ^r , weight 2 ^r -1 and minimum distance 3. The codes have codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes.     

New Linear Codes Over \mathbb{F}_3 and \mathbb{F}_5 and Improvements on Bounds

One of the most important problems of coding theory is to constructcodes with best possible minimum distances. In this paper, we generalize the method introduced by [8] and obtain new codes which improve the best known minimum distance bounds of some linear codes. We have found a new linear ternary code and 8 new linear codes over with improved minimumdistances. First we introduce a generalized version of Gray map,then we give definition of quasi cyclic codes and introduce nearlyquasi cyclic codes. Next, we give the parameters of new codeswith their generator matrices. Finally, we have included twotables which give Hamming weight enumerators of these new codes.     

Codes from Generalized Hexagons

In this paper, we construct some codes that arise from generalized hexagons with small parameters. As our main result we discover two new projective two-weight codes constructed from two-character sets in PG(5,4) and PG(11,2). These in turn are constructed using a new distance-2-ovoid of the classical generalized hexagon H(4). Also the corresponding strongly regular graph is new. The two-character set is the union of two orbits in PG(5,4) under the action of L₂(13). Communicated by: R. Calderbank The first Author is Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O)     

Existence of Perfect 3-Deletion-Correcting Codes

Bours [4] recently showed some constructions for perfect 2 and 3-deletion-correcting codes from combinatorial designs. He settled existence of perfect 2-deletion-correcting codes with words of length 4. However, the existence of perfect 3-deletion-correcting codes with words of length 5, or T^*(2, 5, v), remained unsettled for v 7, 8 (mod 10) and v = 13, 14, 15, 16. In this paper we provide new constructions for these codes from combinatorial designs, and show that a T^*(2, 5, v) exists for all v.     

Construction of nested balanced block designs, rectangular designs andq-ary codes

Construction of nested balanced incomplete block (BIB) designs, nested balanced ternary designs and rectangular designs from given nested BIB designs and resolvable BIB designs are described. New constructions ofq-ary codes from nested BIB designs and balanced bipartite weighing designs are also given.     

Higher Weights for Ternary and Quaternary Self-Dual Codes*

We study higher weights applied to ternary and quaternary self-dual codes. We give lower bounds on the second higher weight and compute the second higher weights for optimal codes of length less than 24. We relate the joint weight enumerator with the higher weight enumerator and use this relationship to produce Gleason theorems. Graded rings of the higher weight enumerators are also determined. This work was supported in part by Northern Advancement Center for Science & Technology and the Natural Sciences and Engineering Research Council of Canada.     

Linear Codes with Non-Uniform Error Correction Capability

This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different code words. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. This is accomplished by a new method called parity sectioned reduction. In this method, the parity check matrix of a uniform error correcting linear code is reduced by dropping some rows and columns and the error range inequalities are modified.