We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for $n in {8, 24, 32, 48, 56, 72, 96}$ every extremal Type I
I code of length $n$ is generated by its codewords of minimal weight. Where Ozeki and Kominers used spherical harmonics and weighted theta functions, we use discrete harmonic polynomials and harmonic weight enumerators. Along we way we introduce $tfrac12$-designs as a discrete analog of Venkovs spherical designs of the same name.
For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.
The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the
fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (-2)-curves on X related to the incidence graph of points and lines of IP^2(IF_4).
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently s
ymmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
Let $n$ be a positive integer. Let $mathbf U$ be the unit disk, $pge 1$ and let $h^p(mathbf U)$ be the Hardy space of harmonic functions. Kresin and Mazya in a recent paper found the representation for the function $H_{n,p}(z)$ in the inequality $$|f
^{(n)} (z)|leq H_{n,p}(z)|Re(f-mathcal P_l)|_{h^p(mathbf U)}, Re fin h^p(mathbf U), zin mathbf U,$$ where $mathcal P_l$ is a polynomial of degree $lle n-1$. We find or represent the sharp constant $C_{p,n}$ in the inequality $H_{n,p}(z)le frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}$. This extends a recent result of the second author and Markovic, where it was considered the case $n=1$ only. As a corollary, an inequality for the modulus of the $n-{th}$ derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves some recent result of Kresin and Mazya.
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean la
ttices. Namely, we use the finite-dimensional representation theory of sl_2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus-Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.
In this note, we give simple examples of sets S of quadratic forms that have minimal S-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh in the negative.
We show that, if L is an extremal Type II lattice of rank 40 or 80, then L is generated by its vectors of norm min(L)+2. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices L are generated by their vectors of norms min(L) and min(L)+2.
We give a new, purely coding-theoretic proof of Kochs criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkovs approach to the classification of the root syste
ms of Type II lattices in R^{24}, and gives a new instance of the analogy between lattices and codes.
