### 数学代写|编码理论代写Coding theory代考|MTH4107

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## 数学代写|编码理论代写Coding theory代考|Constructions of Codes with Prescribed Automorphisms

Huffman and Yorgov (see $[999,1928,1929]$ ) developed a method for constructing binary self-dual codes via an automorphism of odd prime order. Their method was extended by other authors for automorphisms of odd composite order and for automorphisms of order 2 $[272,281,616]$.

Huffman has also studied the properties of the linear codes over $\mathbb{F}{q}$, having an automorphism of prime order $p$ coprime with $q[1000]$. Further, he has continued with Hermitian and additive self-dual codes over $\mathbb{F}{4}[1001,1006]$, and with self-dual codes over rings $[1004,1007]$.
Let $\mathcal{C}$ be a binary self-dual code of length $n$ with an automorphism $\sigma$ of prime order $p \geq 3$ with exactly $c$ independent $p$-cycles and $f=n-c p$ fixed points in its decomposition. We may assume that
$$\sigma=(1,2, \cdots, p)(p+1, p+2, \cdots, 2 p) \cdots((c-1) p+1,(c-1) p+2, \cdots, c p)$$
and say that $\sigma$ is of type $p-(c, f)$. We present the main theorems about the structure of such a code. This structure has been used by many authors in order to construct optimal self-dual codes with different parameters.

Theorem 4.4.20 ([999]) Let $\mathcal{C}$ be a binary $[n, n / 2]$ code with automorphism $\sigma$ from (4.3). Let $\Omega_{1}={1,2, \ldots, p}, \ldots, \Omega_{c}={(c-1) p+1,(c-1) p+2, \ldots, c p}$ denote the cycles of $\sigma$, and let $\Omega_{c+1}={c p+1}, \ldots, \Omega_{c+f}={c p+f=n}$ be the fixed points of $\sigma$. Define
\begin{aligned} &F_{\sigma}(\mathcal{C})={\mathbf{v} \in \mathcal{C} \mid \sigma(\mathbf{v})=\mathbf{v}} \ &E_{\sigma}(\mathcal{C})=\left{\mathbf{v} \in \mathcal{C} \mid \mathbf{w t}{\mathrm{H}}\left(\mathbf{v} \mid \Omega{i}\right) \equiv 0 \quad(\bmod 2), i=1,2, \ldots, c+f\right} \end{aligned}
where $\mathbf{v}{\mid \Omega{i}}$ is the restriction of $\mathbf{v}$ on $\Omega_{i} .$ Then $\mathcal{C}=F_{\sigma}(\mathcal{C}) \oplus E_{\sigma}(\mathcal{C}), \operatorname{dim}\left(F_{\sigma}(\mathcal{C})\right)=\frac{c+f}{2}$, and $\operatorname{dim}\left(E_{\sigma}(\mathcal{C})\right)=\frac{c(p-1)}{2} .$

Theorem 4.4.21 ([1928]) Let $\mathcal{C}$ be a binary $[n, n / 2]$ code with automorphism $\sigma$ from (4.3).
Let $\pi: F_{\sigma}(\mathcal{C}) \rightarrow \mathbb{F}{2}^{c+f}$ be the projection map, where, for $\mathbf{v} \in F{\sigma}(\mathcal{C}),(\pi(\mathbf{v})){i}=v{j}$ for some $j \in \Omega_{i}, i=1,2, \ldots, c+f$. Let $\mathcal{E}$ (respectively $\mathcal{P}$ ) be the set of all even-weight vectors in $\mathbb{F}{2}^{p}$ (respectively even-weight polynomials in $\left.\mathbb{F}{2}[x] /\left\langle x^{p}-1\right\rangle\right)$. Define $\varphi^{\prime}: \mathcal{E} \rightarrow \mathcal{P} b y$ $\varphi^{\prime}\left(v_{0} v_{1} \cdots v_{p-1}\right)=v_{0}+v_{1} x+\cdots+v_{p-1} x^{p-1} .$ Let $E_{\sigma}(\mathcal{C})^{}$ be $E_{\sigma}(\mathcal{C})$ punctured on all the fixed points of $\sigma$. Define $\varphi: E_{\sigma}(\mathcal{C})^{} \rightarrow \mathcal{P}^{c}$ by $\varphi(\mathbf{v})=\left(\varphi^{\prime}\left(\mathbf{v}{\mid \Omega{1}}\right), \varphi^{\prime}\left(\mathbf{v}{\mid \Omega{2}}\right), \ldots, \varphi^{\prime}\left(\mathbf{v} \mid \Omega_{c}\right)\right)$ for $\mathbf{v} \in E_{\sigma}(\mathcal{C})^{} \subseteq \mathcal{E}^{c}$. Then $\mathcal{C}$ is self-dual if and only if the following two conditions hold: (a) $\mathcal{C}{\pi}=\pi\left(F{\sigma}(\mathcal{C})\right)$ is a binary self-dual code of length $c+f$, and
(b) for every two vectors $\mathbf{u}, \mathbf{v} \in \mathcal{C}{\varphi}=\varphi\left(E{\sigma}(\mathcal{C})^{}\right)$, we have $\sum_{i=1}^{c} u_{i}(x) v_{i}\left(x^{-1}\right)=0$ where $u_{i}(x)=\varphi^{\prime}\left(\mathbf{u}{\mid \Omega{i}}\right)$ and $v_{i}(x)=\varphi^{\prime}\left(\mathbf{v}{\mid \Omega{i}}\right)$ for $i=1,2, \ldots, c .$

## 数学代写|编码理论代写Coding theory代考|Enumeration and Classification

Remark 4.5.1 The main tool to classify self-dual codes is based on the so-called mass formula which gives the possibility of checking whether the classification is correct. The number of the self-dual binary codes of even length $n$ is $N(n)=\prod_{i=1}^{n / 2-1}\left(2^{i}+1\right)$. If $\mathcal{C}$ has length $n$, then the number of codes equivalent to $\mathcal{C}$ is $n ! /|\operatorname{PAut}(\mathcal{C})|$. To classify binary selfdual codes of length $n$, it is necessary to find inequivalent self-dual codes $\mathcal{C}{1}, \ldots, \mathcal{C}{r}$ so that the following mass formula holds:
$$N(n)=\sum_{i=1}^{r} \frac{n !}{\left|\operatorname{PAut}\left(\mathcal{C}{i}\right)\right|} .$$ There are such formulas for all families of self-dual and also of self-orthogonal codes. Detailed information is presented in [1008, 1555]. See also Proposition 7.5.1. Theorem 4.5.2 We have the following mass formulas. (a) For self-dual binary codes of even length $n$, $$\sum{j} \frac{n !}{\left|\operatorname{PAut}\left(\mathcal{C}{j}\right)\right|}=\prod{i=1}^{n / 2-1}\left(2^{i}+1\right)$$
(b) For doubly-even self-dual binary codes of length $n \equiv 0(\bmod 8)$,
$$\sum_{j} \frac{n !}{\left|\operatorname{PAut}\left(\mathcal{C}{j}\right)\right|}=\prod{i=1}^{n / 2-2}\left(2^{i}+1\right)$$

(c) For self-dual ternary codes of length $n \equiv 0(\bmod 4)$,
$$\sum_{j} \frac{2^{n} n !}{\left|\operatorname{MAut}\left(\mathcal{C}{j}\right)\right|}=2 \prod{i=1}^{n / 2-1}\left(3^{i}+1\right)$$
(d) For Hermitian self-dual codes over $\mathbb{F}{4}$ of even length $n$, $$\sum{j} \frac{2 \cdot 3^{n} n !}{\left|\Gamma \operatorname{Aut}\left(\mathcal{C}{j}\right)\right|}=\prod{i=1}^{n / 2-1}\left(2^{2 i+1}+1\right)$$
In each case, the summation is over all $j$, where $\left{\mathcal{C}{j}\right}$ is a complete set of representatives of inequivalent codes of the given type. The automorphism group $\operatorname{CAut}\left(\mathcal{C}{j}\right)$ is the set of all semi-linear monomial transformations from $\mathbb{F}{4}^{n}$ to $\mathbb{F}{4}^{n}$ that fix $\mathcal{C}_{j}$; see [1008, Section 1.7].

## 数学代写|编码理论代写Coding theory代考|Designs Supported by Codes

The support of a nonzero vector $\mathbf{x}=x_{1} \cdots x_{n} \in \mathbb{F}{q}^{n}$ is the set of indices of its nonzero coordinates: $\operatorname{supp}(\mathbf{x})=\left{i \mid x{i} \neq 0\right}$

Definition 5.2.1 A design $D$ is supported by a block code $\mathcal{C}$ of length $n$ if the points of $D$ are labeled by the $n$ coordinates of $\mathcal{C}$, and every block of $D$ is the support of some nonzero codeword of $\mathcal{C}$.

Remark 5.2.2 If $\mathcal{C}$ is a linear code over a finite field of order $q>2$, and $\mathbf{c}$ is a codeword of weight $w>0$, all $q-1$ nonzero scalar multiples of $\mathbf{c}$ have the same support. To avoid repeated blocks, we associate only one block with all scalar multiples of c. Suppose that $D$ is a $t-(n, w, \lambda)$ design supported by a linear $q$-ary code $\mathcal{C}$. It follows that the number of blocks $b$ of $D$ is smaller than or equal to $A_{w} /(q-1)$, where $A_{w}$ is the number of codewords of weight $w$. If the support of every codeword of weight $w$ is a block of $D$, then we have and the parameter $\lambda$ can be computed using $(5.2)$ and (5.3):
$$\lambda=\frac{A_{w}}{q-1} \cdot \frac{\left(\begin{array}{c} w \ t \end{array}\right)}{\left(\begin{array}{c} n \ t \end{array}\right)} .$$
Theorem 5.2.3 If a code is invariant under a monomial group that acts t-transitively or $t$-homogeneously on the set of coordinates, the supports of the codewords of any nonzero weight form a t-design.

Corollary 5.2.4 If $\mathcal{C}$ is a cyclic code of length $n$, the supports of all codewords of any nonzero weight $w$ form a 1-design.

## 数学代写|编码理论代写Coding theory代考|Constructions of Codes with Prescribed Automorphisms

σ=(1,2,⋯,p)(p+1,p+2,⋯,2p)⋯((C−1)p+1,(C−1)p+2,⋯,Cp)

\begin{aligned}&F_{\sigma}(\mathcal{C})={\mathbf{v}\in \mathcal{C}\mid \sigma(\mathbf{v})=\mathbf{v}}\ &E_{\sigma}(\mathcal{C})=\left{\mathbf{v} \in \mathcal{C} \mid \mathbf{wt}{\mathrm{H}}\left(\mathbf{v} \mid \Omega{i}\right)\equiv 0\quad(\bmod2),i=1.2,\ldots,c+f\right}\end{aligned}\begin{aligned}&F_{\sigma}(\mathcal{C})={\mathbf{v}\in \mathcal{C}\mid \sigma(\mathbf{v})=\mathbf{v}}\ &E_{\sigma}(\mathcal{C})=\left{\mathbf{v} \in \mathcal{C} \mid \mathbf{wt}{\mathrm{H}}\left(\mathbf{v} \mid \Omega{i}\right)\equiv 0\quad(\bmod2),i=1.2,\ldots,c+f\right}\end{aligned}

(b) 对于每两个向量在,在∈C披=披(和σ(C))， 我们有∑一世=1C在一世(X)在一世(X−1)=0在哪里在一世(X)=披′(在∣Ω一世)和在一世(X)=披′(在∣Ω一世)为了一世=1,2,…,C.

## 数学代写|编码理论代写Coding theory代考|Enumeration and Classification

ñ(n)=∑一世=1rn!|帕特⁡(C一世)|.对于所有自对偶和自正交码族都有这样的公式。详细信息见 [1008, 1555]。另见提案 7.5.1。定理 4.5.2 我们有以下质量公式。(a) 对于偶数长度的自对偶二进制码n,

∑jn!|帕特⁡(Cj)|=∏一世=1n/2−1(2一世+1)
(b) 对于长度的双偶自对偶二进制码n≡0(反对8),

∑jn!|帕特⁡(Cj)|=∏一世=1n/2−2(2一世+1)

(c) 对于长度的自对偶三进制码n≡0(反对4),

∑j2nn!|毛⁡(Cj)|=2∏一世=1n/2−1(3一世+1)
(d) 对于 Hermitian 自对偶码F4等长n,

∑j2⋅3nn!|Γ或者⁡(Cj)|=∏一世=1n/2−1(22一世+1+1)

## 数学代写|编码理论代写Coding theory代考|Designs Supported by Codes

λ=一个在q−1⋅(在 吨)(n 吨).

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