In this paper, we give a representation of the average of complete joint weight enumerators of two linear codes of length n over F and Z in terms of the compositions of n and their distributions in the codes. We also obtain a generalization of the representation for the average of g-fold complete joint weight enumerators of codes over F and Z . Finally, the average of intersection numbers of a pair of Type III (resp. Type IV) codes, and its second moment are found. q k q k

<p>Let Λ be any integral lattice in Euclidean space. It has been shown that for every integer n>0, there is a hypersphere that passes through exactly n points of Λ. Using this result, we introduce new lattice invariants and give some computational results related to two-dimensional Euclidean lattices of class number one.</p>

Let be an extremal binary doubly even self-dual code of length and be the support design of for a weight . We introduce the two numbers and : is the largest integer such that, for all wight, is a -design; denotes the largest integer such that there exists a such that is a -design. In this paper, we consider the possible values of and .

In this paper, we study the congruences for the Fourier coefficients of the Mathieu mock theta function, which appears in the Mathieu moonshine phenomenon discovered by Eguchi, Ooguri, and Tachikawa. (C) 2014 Elsevier Inc. All rights reserved.

Let D be the support design of the minimum weight of an extremal binary doubly even self-dual [24m, 12m, 4m + 4] code. In this note, we consider the case when D becomes a t-design with t &gt;= 6.

In this note, we describe the parity of the coefficients of the McKay-Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan, and Harvey stated in connection with umbral moonshine for the case of Mathieu moonshine.

For lengths 8, 16, and 24, it is known that there is an extremal Type II Z(2k)-code for every positive integer k. In this paper, we show that there is an extremal Type II Z(2k)-code of lengths 32, 40, 48, 56, and 64 for every positive integer k. For length 72, it is also shown that there is an extremal Type II Z(4k)-code for every positive integer k with k &gt;= 2.

In this paper, we define the concept of a spherical T-design for a finite subset of a sphere. Then, we give some examples of such designs using the Z(2)-lattice, which is related to Lehmer's conjecture. (C) 2012 Elsevier B.V. All rights reserved.

In 1947, Lehmer conjectured that the Ramanujan tau-function tau(m) is non-vanishing for all positive integers m, where tau(m) are the Fourier coefficients of the cusp form Delta of weight 12. It is known that Lehmer's conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that tau(m) = 0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V-b)(m+1) is a conformal 12-design. Therefore, Lehmer's conjecture is now reformulated in terms of conformal t-designs. (c) 2012 Elsevier Inc. All rights reserved.

In this paper, we show that there is a frame of norm k in the odd Leech lattice for every k &gt;= 3. (C) 2012 Elsevier Inc. All rights reserved.

Let Lambda be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer n &gt; 0, there is a circle in the plane R-2 that passes through exactly n points of Lambda. (C) 2011 Elsevier Inc. All rights reserved.

In this paper, we study the congruences of the Fourier coefficients of the Mathieu mock theta function, which appears in the Mathieu moonshine phenomenon discovered by Eguchi, Ooguri, and Tachikawa.

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean a-code if any distinct two points in the set are separated at least by a. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean a-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean t-designs as an analogue of the linear programming bound.

It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.

We use the theory of algebraic fields to establish that none of the shells of the lattice Z(2) (resp. A(2)) is a 4-design (resp. 6-design). We discuss the connection between spherical designs and imaginary quadratic fields.

In this paper we give a new upper bound on the minimum Euclidean weight of Type II Z(2k)-codes and the concept of extremality for the Euclidean weights when k = 3 4 5 6 Together with the known result we demonstrate that there is an extremal Type II Z(2k)-code of length 8m (m &lt;= 8) when k = 3,4,5 6 (C) 2010 Elsevier Inc All rights reserved

In 1947, Lehmer conjectured that the Ramanujan tau-function tau(m) never vanishes for all positive integers m, where tau(m) are the Fourier coefficients of the cusp form Delta(24) of weight 12. Lehmer verified the conjecture in 1947 for m &lt; 214928639999. In 1973, Serre verified up to m &lt; 10(15), and in 1999, Jordan and Kelly for m &lt; 22689242781695999.
The theory of spherical t-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's tau-function gives the coefficients of a weighted theta series of the E-8-lattice. It is shown, by Venkov, de la Harpe, and Pache, that tau(m) = 0 is equivalent to the fact that the shell of norm 2m of the E-8-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t-design.
Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m-th Fourier coefficient of the weighted theta series of the Z(2)-lattice and the A(2)-lattice does not vanish, when the shell of norm in of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z(2)-lattice (resp. A(2)-lattice).

For any McKay-Thompson series which appear in Moonshine, the Hecke-type Faber polynomial dagger P-n (X) of degree n is defined. The Hecke-type Faber polynomials are of course special cases of the Faber polynomials introduced by Faber a century ago. We first study the locations of the zeros of the Hecke-type Faber polynomials of the 171 monstrous types, as well as those of the 157 non-monstrous types. We have calculated, using a computer, the zeros for all n &lt;= 50. These results suggest that in many (about 13%) of the cases, we can expect that all of the zeros of P-n (x) are real numbers. In particular, we prove rigorously that the zeros of the Hecke-type Faber polynomials (of any degree) for the McKay-Thompson series of type 2A are real numbers. We also discuss the effect of the existence of harmonics, and the effect of a so-called dash operator. We remark that by the dash operators, we obtain many replicable functions (with rational integer coefficients) which are not necessarily completely replicable functions. Finally, we study more closely the curves on which the zeros of the Hecke-type Faber polynomials for type 5B lie in particular in connection with the fundamental domain (on the upper half plane) of the group Gamma(0)(5), which was studied by Shigezumi and Tsutsumi. At the end, we conclude this paper by stating several observations and speculations.