W. Cary Huffman

Fundamentals of Error-Correcting Codes: Fundamentals of Error-Correcting Codes

Preface 1. Basic concepts of linear codes 2. Bounds on size of codes 3. Finite fields 4. Cyclic codes 5. BCH and Reed-Soloman codes 6. Duadic codes 7. Weight distributions 8. Designs 9. Self-dual codes 10. Some favourite self-dual codes 11. Covering radius and cosets 12. Codes over Z4 13. Codes from algebraic geometry 14. Convolutional codes 15. Soft decision and iterative decoding Bibliography Index.

Fundamentals of Error-Correcting Codes: Contents

Preface 1. Basic concepts of linear codes 2. Bounds on size of codes 3. Finite fields 4. Cyclic codes 5. BCH and Reed-Soloman codes 6. Duadic codes 7. Weight distributions 8. Designs 9. Self-dual codes 10. Some favourite self-dual codes 11. Covering radius and cosets 12. Codes over Z4 13. Codes from algebraic geometry 14. Convolutional codes 15. Soft decision and iterative decoding Bibliography Index.

Multipliers and generalized multipliers of cyclic objects and cyclic codes

Abstract In (European J. Combin. Theory 8 (1987), 35–43) Palfy answers the question: Under what conditions on n is it true that two equivalent objects in any class of cyclic combinatorial objects on n elements implies the objects are equivalent, using one of the ϕ(n) multipliers i → ai mod, n, with gcd(a, n) = 1. Palfy proved that this is true precisely when n = 4 or gcd(n, ϕ(n)) = 1. For any odd prime p, we prove that two equivalent objects in any class of cyclic combinatorial objects on n = p2 elements are equivalent using a permutation from a list of no more then ϕ(n) = p(p − 1) permutations. We introduce permutations called generalized multipliers, and we show that two permutation equivalent cyclic codes of length p2 are equivalent by a generalized multiplier times a multiplier. We also develop properties of generalized multipliers and generalized affine maps when n = pm, show that they map cyclic codes to cyclic codes, and show that certain of these maps are in the automorphism group of a cyclic code.

The existence of extremal self-dual [50,25,10] codes and quasi-symmetric 2-(49,9,6) designs

All extremal binary self-dual [50,25,10] codes with an automorphism of order 7 are enumerated. Up to equivalence, there are four such codes, three with full automorphism group of order 21, and one code with full group of order 7. The minimum weight codewords yield quasi-symmetric 2-(49,9,6) designs.

Topological invariants of 2-designs arising from difference families

Hefftner, White, Alpert, and others observed a connection between topology and certain block designs with parameters k = 3 and λ = 2. In this paper the connection is extended to include all values of λ. The topology is also exploited further to produce some new invariants of designs. The topology also gives an upper bound for the order of the automorphism group of the designs studied which leads to a generalization of the Bays-Lambossy theorem. Methods for constructing block designs are also given showing that the results apply and are useful for a large class of designs.

Cyclic double struck capital F q-linear double struck capital F q t -codes

In Calderbank, self-orthogonal additive codes over double struck capital F 4 under the traceinner product were connected to binary quantum codes; a similar connection was given in the non-binary case in Rains. In this paper, we consider a natural generalisation of additive codes to double struck capital F q-linear double struck capital F q t-codes. We develop the theory of these codes when they are cyclic and count them. Then, we place two different trace inner products on these codes and decide precisely when the cyclic ones are self-orthogonal under these two inner products. We examine the case t = 2 in detail giving specific bases for the self-orthogonal and self-dual cyclic codes. When t ≥ 2, we present counts for the number of self-orthogonal and self-dual cyclic codes under each of the inner products.

The biweight enumerator of self-orthogonal binary codes

Abstract Let C be a binary code of length n and let J C ( a , b , c , d ) be its biweight enumerator. If n is even and C is self-dual, then J C is an element of the ring R G of absolute invariants of a certain group G . Under the additional assumption that all codewords of C have weight divisible by 4, a similar result holds with a different group. If n is odd and C is maximal self-orthogonal, then J C is an element of a certain R G -module. Again a similar result holds if the codewords of C have weights divisible by 4. The groups involved are related to finite groups generated by reflections. In this paper the structure of these groups is described, and polynomial bases for the rings and modules in question are obtained. This answers a question posed in The Theory of Error- correcting Codes by F.J. MacWilliams and N.J.A. Sloane.

On the equivalence of codes and codes with an automorphism having two cycles

Abstract In this paper we generalize a result of V.Y. Yorgov giving sufficient conditions under which two codes are equivalent. We then use this result to construct and to count the number of inequivalent \ s [2 r , r \ s ] and \ s [2 r +2, r +1\ s ] self-dual codes, over an arbitrary field, with an automorphism of prime order r .

Constructions and topological invariants of 2-(n,3,l) designs with group actions

Connections between 2-(ν3, λ) designs and topology are exploited to produce topological invariants of these designs. When these designs have a natural group action with 0 or 1 fixed point, these invariants are easily computed. Methods of constructing 2-(ν, 3, λ) designs with such a group action are given. Examples of this construction are given and the invariants are used to distinguish the designs.

Invariants and constructions of mendelsohn designs

In this paper we examine Mendelsohn designs and some connections to topology which lead to an easily described algorithm for computing invariants of these designs. The results are applied to designs which have natural group actions. We also use the topology to describe when ordinary two-fold triple systems with a group action lead to Mendelsohn designs with the same group action. Procedures for constructing Mendelsohn designs are also given. In particular, we give necessary and sufficient conditions for constructing 2-(v, 4, 1) Mendelsohn designs.