Michio Ozeki

On the covering radius of Z/sub 4/-codes and their lattices

In this correspondence, we investigate the covering radius of codes over Z/sub 4/ for the Lee and Euclidean distances in relation with those of binary nonlinear codes and lattices obtained by the Gray map and Construction A/sub 4/, respectively. We give several upper and lower bounds on covering radii, including Z/sub 4/-analogs of the sphere-covering bound, the packing radius bound, the Delsarte bound, and the redundancy bound. We show that any Euclidean-optimal Type II code of length 24 has covering radius 8 with respect to the Euclidean distance. We determine the covering radius of the Klemm codes with respect to the Lee distance. We derive lower bounds on the covering radii of the Niemeier lattices.

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over \Bbb Z 6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ℤ6-codes. Then extremal self-dual ℤ6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.

Extremal self-dual codes with the smallest covering radius

Abstract In this note, an extremal doubly-even [40,20,8] code with covering radius 7 is given. This code is the first extremal doubly-even code whose covering radius does not meet the Delsarte bound. Extremal singly-even codes with the smallest covering radius are also given. One of them is an extremal singly-even [44,22,8] code with covering radius 7, which determines t [43,22]=6 by puncturing.

On covering radii and coset weight distributions of extremal binary self-dual codes of length 40

In the present paper we develop a method to determine the coset weight distributions and covering radius of doubly even self-dual extremal binary codes of length 40. The method is algebraic in nature and largely eliminates necessary computations by electronic computers. The method easily applies to longer codes (e.g. self-dual [56,28,12] binary codes) or to non-extremal codes.

Jacobi polynomials for singly even self-dual codes and the covering radius problems

In this correspondence, we develop a method to determine the complete coset weight distributions of the class of singly even self-dual binary codes. Our basic tool is the Jacobi polynomials for the code. It describes and controls the coset weight enumerators. As the results of our present method, we give the complete coset weight distributions of some extremal singly even self-dual codes of lengths 14, 22, 32, 36, and 40, respectively. We give the generator matrices of the used codes of lengths 36 and 40, respectively.

On the Ring of Simultaneous Invariants for the Gleason-MacWilliams Group

We construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason?MacWilliams group. We find this canonical set in the vector space (?i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants Cx, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (?i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms.

On the covering radius problem for ternary self-dual codes

In the present paper we develop a method to determine the cost weight distributions and covering radius of extremal ternary self-dual codes of various lengths. The notion of modified Jacobi polynomials is introduced and elementary aspects of it art discussed. Algebraic foundation has much room for further investigations. Since the algorithms underlining in our approach to this problem are sound, and we can bet many numerical results as far as the computer runs in a reasonable time.

On covering radii and coset weight distributions of extremal binary self-dual codes of length 56

We present a method to determine the complete coset weight distributions of doubly even binary self-dual extremal [56, 28, 12] codes. The most important steps are (1) to describe the shape of the basis for the linear space of rigid Jacobi polynomials associated with such codes in each index i, (2) to describe the basis polynomials for the coset weight enumerators of the assigned coset weight i by means of rigid Jacobi polynomials of index i. The multiplicity of the cosets of weight i have a connection with the frequency of the rigid reference binary vectors v of weight i for the Jacobi polynomials. This information is sufficient to determine the complete coset weight distributions. Determination of the covering radius of the codes is an immediate consequence of this method. One important practical advantage of this method is that it is enough to get information on 8190 codewords of weight 12 (minimal-weight words) in each such code for computing every necessary information.

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.

Notes on the shadow process in self-dual codes

In the present paper we present some tools in exploring the shadow process in self-dual codes for the purpose of finding singly even self-dual binary codes with higher minimal distances. The present paper can be viewed as a supplementary work to the preceding works (IEEE Trans. Inform. Theory 36 (1990) 1319; IEEE Trans. Inform Theory 37 (1991) 1222).