Boris Venkov

The nonexistence of certain tight spherical designs

In this paper, the nonexistence of tight spherical designs is shown in some cases left open to date. Tight spherical 5-designs may exist in dimension n = (2m + 1)2 − 2, and the existence is known only for m = 1, 2. In the paper, the existence is ruled out under a certain arithmetic condition on the integer m, satisfied by infinitely many values of m, including m = 4. Also, nonexistence is shown for m = 3. Tight spherical 7-designs may exist in dimension n = 3d2 − 4, and the existence is known only for d = 2, 3. In the paper, the existence is ruled out under a certain arithmetic condition on d, satisfied by infinitely many values of d, including d = 4. Also, nonexistence is shown for d = 5. The fact that the arithmetic conditions on m for 5-designs and on d for 7-designs are satisfied by infinitely many values of m and d, respectively, is shown in the Appendix written by Y.-F. S. Pétermann. §

On tight spherical designs

Let X be a tight t-design of dimension n for one of the open cases t=5 or t=7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely many values for n.

Construction of spherical cubature formulas using lattices

We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming.

Unimodular lattices with long shadow

Let L be an odd unimodular lattice of dimension n with shadow n−16. If min(L)⩾3 then dim(L)⩽46 and there is a unique such lattice in dimension 46 and no lattices in dimensions 44 and 45. To prove this, a shadow theory for theta series with spherical coefficients is developed.

Classification of Ternary Extremal Self-Dual Codes of Length 28

All 28-dimensional unimodular lattices with minimum norm 3 are known. Using this classification, we give a classification of ternary extremal self-dual codes of length 28. Up to equivalence, there are 6,931 such codes.

On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

On integral lattices having an odd minimum

The kissing number of integral lattices of odd minimum is studied, with special emphasis on the case of minimum 3.

Séries de croissance et séries d'Ehrhart associées aux réseaux de racines

Let K be an integral convex polytope in R″ of which the integral points generate Z″: one defines on one hand the usual growth series of the group Z″ with respect to K ∩ Z″ and on the other hand the Ehrhart series of which the k-th coefficient counts points in (kK) ∩ Z″. When K is the convex hull of a root system, it is remarkable that these series coincide up to a factor 11−z. This follows from explicit computations which are the main result of this Note.

Odd unimodular lattices

One investigates the relation between even and odd unimodular lattices in ℝn. One constructs the operation of taking even associates for odd lattices. One computes the action of this operation upon the theta-series. One investigates the even associates of lattices of the form Λ⊕ℤs.

LOW-DIMENSIONAL STRONGLY PERFECT LATTICES. III: DUAL STRONGLY PERFECT LATTICES OF DIMENSION 14

The extremal 3-modular lattice