Jeffrey S. Leon

A probabilistic algorithm for computing minimum weights of large error-correcting codes

An algorithm is developed that can be used to find, with a very low probability of error (10/sup -100/ or less in many cases), the minimum weights of codes far too large to be treated by any known exact algorithm. The probabilistic method is used to find minimum weights of all extended quadratic residue codes of length 440 or less. The probabilistic algorithm is presented for binary codes, but it can be generalized to codes over GF(q) with q>2. >

Duadic Codes

A new family of binary cyclic (n,(n + 1)/2) and (n,(n - 1)/2) codes are introduced, which include quadratic residue (QR) codes when n is prime. These codes are defined in terms of their idempotent generators, and they exist for all odd n = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{r}^{a_{r}} where each p_{i} is a prime \equiv \pm 1 \pmod{8} . Dual codes are identified. The minimum odd weight of a duadic (n,(n + 1)/2) code satisfies a square root bound. When equality holds in the sharper form of this bound, vectors of minimum weight hold a projective plane. The unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent (73,37,9) duadic codes. All duadic codes of length less than 127 are identified, and the minimum weights of their extensions are given. One of the duadic codes of length 113 has greater minimum weight than the QR code of that length.

On an algorithm for finding a base and a strong generating set for a group given by generating permutations

This paper deals with the problem of finding a base and strong generating set for the group generated by a given set of permutations. The concepts of base and strong generating set were introduced by Sims (51, (61 and provide the most effective tool for computing with permutation groups of high degree. One algorithm, originally proposed by Sims [71, is described in detail; its behavior on a number of groups is studied, and the influence of certain parameters on its performance is investigated. Another algorithm, developed by the author, is given, and it is shown how the two algorithms may be combined to yield an exceptionally fast and

Permutation group algorithms based on partitions, I: Theory and algorithms

A technique for computing in permutation groups of high degree is developed. The technique uses the idea of successive refinement of ordered partitions, introduced by B. McKay in connection with the graph isomorphism problem, to supplement the techniques of base and strong generating set developed earlier by Sims. Applications to a number of specific problems in computational group theory are presented.

Self-Dual Codes over GF(5)

Abstract It is shown that a self-orthogonal code over GF(5) which is generated by words of weight 4 has a decomposition into components belonging to three infinite families: dn (n = 4, 5, 6, 7, 8, 10, 12,…), en (n = 6, 8, 10,…) and Fn (n = 6, 8, 10,…). All maximal self-orthogonal (and self-dual) codes of length ⩽ 12 are classified, and a number of interesting codes of greater length are constructed.

Computing automorphism groups of error-correcting codes

An algorithm is described for computing the automorphism group of an error correcting code. The algorithm determines the order of the automorphism group and produces a set of monomial permutations which generate the group. It has been implemented on a computer and has been used successfully on a great number of codes of moderate length.

All self-dual Z/sub 4/ codes of length 15 or less are known

We classify the self-dual Z/sub 4/ codes of all lengths from 10 through 15, extending the previous classification through length 9.

All Z4Codes of Type II and Length 16 Are Known

We show that there are 133 inequivalent Type II codes over Z4of length 16. We give the number of each type 4i·2j, where 2i+j=16, a generator matrix for each code, the order of its automorphism group, and its minimum Lee weight. A (partial) symmetrized weight enumerator of each code is given. The highest minimum Lee weight amongst these codes is 8, and there are 5 inequivalent codes attaining this highest Lee weight. A new computer algorithm to determine the automorphism group of a Z4code was used

On weights in duadic codes

Abstract In [3] we introduced a new family of binary, cyclic ( n, (n+1) 2 ) and ( n, (n-1) 2 ) codes which include quadratic residue (Q. R.) codes when n is prime. These codes are defined in terms of their idempotent generators and they exist for all n=p1a1p2a2…prar where each pi is a prime ≡ ±1 (mod 8). This family of codes has properties analogous to properties of Q.R. codes, and we investigate some of these in this paper. In particular, we determine conditions on n under which one or every extended duadic code is self-dual or its dual is the other associated duadic code. In the self-dual case we characterize weights and strengthen the square root bound. In addition, we show that every extended cyclic, self-dual binary code is a duadic code. Hence we know at which lengths these codes exist and that certain Reed-Muller codes are duadic codes. In (Leon, Masley, and Pless, IEEE. Trans. Inform. Theory IT 30 (1984), 709–714) information was given about duadic codes until length 119. Here we give extensive tables describing minimum weights and duals of all duadic codes until length 241. Several duadic codes are better than the Q.R. codes of the same length, and one is the best code known of its length.

Classification of 3-(24, 12, 5) designs and 24-dimensional Hadamard matrices

Recently the authors completed the classification of 3-(24, 12, 5) designs up to isomorphism. These designs are closely related to 24-dimensional Hadamard matrices, and the work on designs leads to a classification of the matrices up to equivalence. Hadamard matrices of lower dimension had been determined previously by Hall [2-41. This article described the techniques used in the classification and contains a summary of the results. It provides information from which each of the 129 designs and 59 matrices can be constructed, and it gives the order and orbit lengths of the automorphism group of each design and matrix. An explicit list of the matrices, together with generating permutations for their automorphism groups, may be found in [5]. The corresponding data for designs appear in 161. In general, a 3-(413. + 4, 21 + 2,L) design D = (P, B) is called a Hadamard design (P and B denote the sets of points and blocks, respectively). In such a design the set of 8A + 6 blocks is necessarily closed under complementation [7]; thus, if 6 denotes P a, B consists of 4,l + 3 block pairs 01* = {a, C}, which we term parallel classes. For a E P, we let B, denote the set of blocks containing the point a.