Related papers: A note on Assmus--Mattson type theorems
Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we…
We previously proposed the first nontrivial examples of a code having support $t$-designs for all weights obtained from the Assmus-Mattson theorem and having support $t'$-designs for some weights with some $t'>t$. This suggests the…
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 \geq 6.
By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs…
Special functions, coding theory and $t$-designs have close connections and interesting interplay. A standard approach to constructing $t$-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the…
In the present paper, we give the Assmus--Mattson type theorem for near-extremal Type I and even formally self-dual codes. We show the existence of $1$-designs or $2$-designs for these codes. As a corollary, we prove the uniqueness of a…
Let $C$ be a two and three-weight ternary code. Furthermore, we assume that $C_\ell$ are $t$-designs for all $\ell$ by the Assmus--Mattson theorem. We show that $t \leq 5$. As a corollary, we provide a new characterization of the (extended)…
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with $s$ classes). This in particular generalizes the Assmus-Mattson-type theorems for…
Coding theory and $t$-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight…
In this paper, we present examples of codes all of whose weight classes support 1-designs, with duals whose classes include two that support 2-designs. We can find these examples in the triply even binary codes of length 48, which have been…
New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$…
Combinatorial designs are closely related to linear codes. In recent year, there are a lot of $t$-designs constructed from certain linear codes. In this paper, we aim to construct $2$-designs from binary three-weight codes. For any binary…
It is shown that an n-dimensional unimodular lattice has minimal norm at most 2[n/24] +2, unless n = 23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the…
We express the weight enumerators of self-dual and doubly even (Type II for short) codes of length $24$ with a specified basis. As a consequence, we present some congruence relations among the weight enumerators.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear…
In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a…
Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the…
For lengths up to 47 except 37, we determine the largest minimum Euclidean weight among all Type I Z4-codes of that length. We also give the first example of an optimal odd unimodular lattice in dimension 41 explicitly, which is constructed…
It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary…
Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a $t$-design. Till now only a small amount of work on constructing $t$-designs…