An upper bound of the value of $$t$$ t of the support $$t$$ t -designs of extremal binary doubly even self-dual codes

Abstract

Let C be an extremal Type III or IV code and D w be the support design of C for a weight w. We introduce the two numbers δ(C) and s(C): δ(C) is the largest integer t such that, for all wight, D w is a t-design; s(C) denotes the largest integer t such that there exists a w such that D w is a t-design. In the present paper, we consider the possible values of δ(C) and s(C).

Design-theoretic analogies between codes, lattices, and vertex operator algebras

“…Then, based on [26, Lemma 31], there exists P ∈ Harm 4 (R 24 ) such that ϑ L,P (z) = d 2 (P )E 4 (z)∆(z), where d 2 (P ) is a non-zero constant. Therefore, based on Theorem 2.7, there exists v P ∈ (V L ) 4 such that Z V L (v P , z) = d 2 (P )E 4 (z)∆(z)/η(z) 24 = q −1 ∞ i=1 c(i)q i . We have d 2 (P ) × c(1) = 0, that is, (V L ) 1 is not a conformal 4-design.…”

A note on Assmus--Mattson type theorems

“…In [22,23], the first and third authors considered the possible occurrence of δ(C) < s(C). By Theorems 1.1 (1) and 1.4, we have…”

On Eisenstein polynomials and zeta polynomials II

“…Then the result follows. The elements of G III are listed on the homepage of one of the authors [12].…”

“…Proof of Theorem 2.2(3). By a direct calculation, Then the result follows.The elements of G IV are listed on the homepage of one of the authors[12].B Proof of Theorem 2.3 for the cases Type III and Type IVProof of Theorem 2.3(2). Let N j T j .…”