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Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$-isometry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $\theta$-twisted $V_{L}$-module. In this paper (Part 1), we show that when the rank of $L$ is $1$, every non-zero weak $V_{L}^{+}$-module contains a non-zero $M(1)^{+}$-module, where $M(1)^{+}$ is the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$., Comment: 29 pages. To appear in Journal of Algebra. We divide the article arXiv:1910.07126 into 3 parts for publication. This is the first part

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $\omega$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $\omega_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s} u\in{\mathbb C}^{\times} u$, and $\omega_i u=0$ for all $i>s$., Comment: 21 pages. arXiv admin note: text overlap with arXiv:1608.07890

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $\omega_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module., Comment: 20 pages; typos corrected

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.

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g∈G and nonnegative rational number n∈ Z /|g| , an associative algebra Ag,n(V) plays an important role in the theory of vertex operator algebras, but the given product in Ag,n(V) depends on the eigenspaces of g. We show that if V has no negative weights then there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra AG,n(V) which covers all Ag,n(V). Let V be a simple vertex operator algebra with no negative weights and let S be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra A α (G, S ) for a suitable 2-cocycle naturally determined by the G-action on S . We show that a duality theorem of Schur–Weyl type holds for the actions of A α (G, S ) and VG on the direct sum of twisted V-modules in S as an application of the theory of AG,n(V). It follows as a natural consequence of the result that for any g∈G every irreducible g-twisted V-module is a completely reducible VG-module.

We construct new extremal self-dual [40,20,8] codes with covering radius 7. It is also shown that the vectors of a fixed weight in a coset of weight 4n+2 in an extremal doubly even self-dual code of length 24n+16 such that the coset has no vector of weight 4n+4 form a 1-design.

C. Bachoc gave a new proof of the Assmus–Mattson theorem for linear binary codes using harmonic weight enumerators which she defined B. We give a new proof of the Assmus–Mattson theorem for linear codes over any finite field using similar methods.

Let Y be any commutative association scheme and we fix any vertex x of Y. Terwilleger introduced a non-commutative, associative, and semi-simple {\bf C}-algebraT\,{=}\,T(x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra ofY with respect to x. Let W (\subset {\bf C}^{|X|}) be an irreducible T(x)-module. W is said to be thin if W satisfies a certain simple condition.Y is said to be thin with respect to x if each irreducible T(x) -module is thin. Y is said to be thin if Y is thin with respect to each vertex in X.The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphsK_4 . Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible T(x)-modules of Doob schemes.

We introduce a notion of a (V,T)-module over a vertex algebra V for an arbitrary positive integer T, which is a generalization of a twisted V-module. Under some conditions on V, we construct an associative algebra A^{T}_{m}(V) for m\in(1/T)\N and an A^{T}_{m}(V)-A^{T}_{n}(V)-bimodule A^{T}_{n,m}(V) for n,m\in(1/T)\N and we establish a one-to-one correspondence between the set of isomorphism classes of simple left A^{T}_{0}(V)-modules and that of simple (1/T)\N-graded (V,T)-modules., Comment: 44 pages, List of Notations added

This is a summary of our recent work. We study a fixed-point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. The classification of the irreducible modules, together with the rationality and the C2-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.