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# 数学代写|组合数学代写Combinatorial Mathematics代考|MATH4306 The Assmus–Mattson theorem and its extensions

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## 数学代写|组合数学代写Combinatorial Mathematics代考|The Assmus–Mattson theorem and its extensions

We introduce a famous theorem by Assmus and Mattson on a construction of designs from codes. Combinatorial 5-designs, which are not directly related to 5-transitive Mathieu groups, were first found by this method [6]. Let $N={1,2, \ldots, n}$ and $F_2={0,1}$. Each element of $F_2^n$ corresponds to a subset of $N$ as follows. For $\boldsymbol{u}=\left(u_1, u_2, \ldots, u_n\right) \in F_2^n$, the subset $\left{i \mid u_i=1,1 \leq i \leq n\right}$ of $N$ is called the support of $\boldsymbol{u}$ and is denoted by $\overline{\boldsymbol{u}}$. If $\boldsymbol{u}$ has weight $m$, then $\overline{\boldsymbol{u}}$ is an $m$-subset of $N$. By this correspondence, the set of codewords in $C$ of weight $m$ is identified with a subset of the set $N^{(m)}$ consisting of $m$-subsets of $N$. This means, in the language of association schemes, a subset in the Hamming scheme $H(n, 2)$ can be described in terms of the Johnson scheme $J(n, m)$.

Theorem 4.1 (Assmus and Mattson (1969) [6]). Let $\mathrm{C}$ be an $[n, k, \delta]$ code over $F_2$. Name$l y, C$ is a $k$-dimensional subspace of $F_2^n$ with minimum distance $\delta$. Let $t<\delta$. Moreover suppose the following condition holds.

There exist at most $\delta$ – $t$ positive integers in ${1,2, \ldots, n-t}$ which arise as the weights of codewords in $C^{\perp}$.
Then the following (1), (2) hold:
(1) $\left{\overline{\boldsymbol{u}} \in N^{(m)} \mid \boldsymbol{u} \in C^{\perp}, w(\boldsymbol{u})=m\right}$ forms a t-design in the Johnson scheme J(n,m);
(2) $\left{\overline{\boldsymbol{u}} \in N^{(m)} \mid \boldsymbol{u} \in C, w(\boldsymbol{u})=m\right}$ forms a t-design in the Johnson scheme J(n,m).
In (1), (2), we only consider the case that there exists a codeword such that $w(\boldsymbol{u})=m$.

## 数学代写|组合数学代写Combinatorial Mathematics代考|t-Designs in regular semilattices

First, we give definitions and basic facts on semilattices.
Definition $4.13$ (Poset). A partial order on a set $L$ is a binary relation $\leq$ on $L$ satisfying the following (1)-(3):
(1) reflexivity: $a \leq a$;
(2) transitivity: if $a \leq b$ and $b \leq c$, then $a \leq c$;
(3) antisymmetry: if $a \leq b$ and $b \leq a$, then $a=b$;
where $a, b, c \in L$. If a binary relation $\leq$ is a partial order on $L$, a pair $(L, \leq)$ is called a partially ordered set or a poset.

Definition 4.14 (Meet semilattice). Let $L$ be a poset. For a pair $a, b$ in $L$, an element of $L$, denoted by $a \wedge b$, is called the meet of $a$ and $b$ if it satisfies the following (1), (2):

(1) $a \wedge b \leq a, a \wedge b \leq b$;
(2) if $c \leq a$ and $c \leq b$ for $c \in L$, then $c \leq a \wedge b$.
Note that the meet $a \wedge b$ is uniquely determined if it exists. The poset $L$ is called a meet semilattice if the meet exists for every pair of elements of $L$.

Definition 4.15 (Join semilattice). Let $L$ be a poset. For a pair $a, b$ in $L$, an element of $L$, denoted by $a \vee b$, is called the join of $a$ and $b$ if it satisfies the following (1), (2):
(1) $a \vee b \geq a, a \vee b \geq b$
(2) if $c \geq a$ and $c \geq b$ for $c \in L$, then $c \geq a \vee b$.
Note that the join $a \vee b$ is uniquely determined if it exists. The poset $L$ is called a join semilattice if the join exists for every pair of elements of $L$. The poset $L$ is called a lattice if it is a meet lattice and a join lattice.

In what follows, a semilattice means a meet semilattice. Assume that for a semilattice $L$, there exists a unique element $u$ such that $u \leq x$ for all $x$. We denote this element by 0 .

## 数学代写|组合数学代写Combinatorial Mathematics代考|The Assmus-Mattson theorem and its extensions

$\boldsymbol{u}=\left(u_1, u_2, \ldots, u_n\right) \in F_2^n$, 子集 $\backslash$ left 缺少或无法识别的分隔符

$m$ ，然右 $\bar{u}$ 是一个 $m$-子集 $N$. 通过这种对应，代码字集在 $C$ 重量 $m$ 被识别为集合的一个子集 $N^{(m)}$ 包含由…组成 $m$-子集的 $N$. 这 意味着，在关联方案的语言中，汉明方客中的一个子集 $H(n, 2)$ 可以用约翰逊方客来苗述 $J(n, m)$.

(1) \left 缺少或无法识别的分隔符 在 Johnson 方案 $\mathrm{J}(\mathrm{n}, \mathrm{m})$ 中形成 $\mathrm{t}$ 设计；
(2) \left 缺少或无法识别的分隔符 在 Johnson 方案 $\mathrm{J}(\mathrm{n}, \mathrm{m})$ 中形成 $\mathrm{t}$ 设计。

## 数学代写|组合数学代写Combinatorial Mathematics代考|t-Designs in regular semilattices

1）自反性: $a \leq a$ ；
（2）传递性: if $a \leq b$ 和 $b \leq c$ ，然后 $a \leq c$;
（3）反对称性: 如果 $a \leq b$ 和 $b \leq a$ ，然后 $a=b$;

(2) :
(1) $a \wedge b \leq a, a \wedge b \leq b$;
（2）如果 $c \leq a$ 和 $c \leq b$ 为了 $c \in L$ ，然后 $c \leq a \wedge b$.

(2) : (
1) $a \vee b \geq a, a \vee b \geq b$
(2) 如果 $c \geq a$ 和 $c \geq b$ 为了c $c \in L$ ，然后 $c \geq a \vee b$.

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