### 数学代写|编码理论代写Coding theory代考|COMP2610

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## 数学代写|编码理论代写Coding theory代考|Puncturing, Extending, and Shortening Codes

There are several methods to obtain a longer or shorter code from a given code; while this can be done for both linear and nonlinear codes, we focus on linear ones. Two codes can be combined into a single code, for example as described in Section 1.11.

Definition 1.7.1 Let $\mathcal{C}$ be an $[n, k, d]{q}$ linear code with generator matrix $G$ and parity check matrix $H$. (a) For some $i$ with $1 \leq i \leq n$, let $\mathcal{C}^{}$ be the codewords of $\mathcal{C}$ with the $i^{\text {th }}$ component deleted. The resulting code, called a punctured code, is an $\left[n-1, k^{}, d^{}\right]$ code. If $d>1, k^{}=k$, and $d^{}=d$ unless $\mathcal{C}$ has a minimum weight codeword that is nonzero on coordinate $i$, in which case $d^{}=d-1$. If $d=1, k^{}=k$ and $d^{}=1$ unless $\mathcal{C}$ has a weight 1 codeword that is nonzero on coordinate $i$, in which case $k^{}=k-1$ and $d^{} \geq 1$ as long as $\mathcal{C}^{}$ is nonzero. A generator matrix for $\mathcal{C}^{}$ is obtained from $G$ by deleting column $i$; $G^{}$ will have dependent rows if $d^{}=1$ and $k^{*}=k-1$. Puncturing is often done on multiple coordinates in an analogous manner, one coordinate at a time.
(b) Define $\widehat{\mathcal{C}}=\left{c{1} c_{2} \cdots c_{n+1} \in \mathbb{F}{q}^{n+1} \mid c{1} c_{2} \cdots c_{n} \in \mathcal{C}\right.$ where $\left.\sum_{i=1}^{n+1} c_{i}=0\right}$, called the extended code. This is an $[n+1, k, \widehat{d}]_{q}$ code where $\hat{d}=d$ or $d+1$. A generator

matrix $\widehat{G}$ for $\widehat{\mathcal{C}}$ is obtained by adding a column on the right of $G$ so that every row sum in this $k \times(n+1)$ matrix is 0. A parity check matrix $\hat{H}$ for $\widehat{\mathcal{C}}$ is
$$\hat{H}=\left[\begin{array}{ccc|c} 1 & \cdots & 1 & 1 \ \hline & & 0 \ & H & & \vdots \ & & & 0 \end{array}\right]$$
(c) Let $S$ be any set of $s$ coordinates. Let $\mathcal{C}(S)$ be all codewords in $\mathcal{C}$ that are zero on $S$. Puncturing $\mathcal{C}(S)$ on $S$ results in the $\left[n-s, k_{S}, d_{S}\right]{q}$ shortened code $\mathcal{C}{S}$ where $d_{S} \geq d$. If $\mathcal{C}^{\perp}$ has minimum weight $d^{\perp}$ and $s<d^{\perp}$, then $k_{S}=k-s$.

## 数学代写|编码理论代写Coding theory代考|Equivalence and Automorphisms

Two vector spaces over $\mathbb{F}_{q}$ are considered the same (that is, isomorphic) if there is a nonsingular linear transformation from one to the other. For linear codes to be considered the same, we want these linear transformations to also preserve weights of codewords. In Theorem 1.8.6, we will see that these weight preserving linear transformations are directly related to monomial matrices. This leads to two different concepts of code equivalence for linear codes.

Definition 1.8.1 If $P \in \mathbb{F}{q}^{n \times n}$ has exactly one 1 in each row and column and 0 elsewhere, $P$ is a permutation matrix. If $M \in \mathbb{F}{q}^{n \times n}$ has exactly one nonzero entry in each row and column, $M$ is a monomial matrix. If $\mathcal{C}$ is a code over $\mathbb{F}{q}$ of length $n$ and $A \in$ $\mathbb{F}{q}^{n \times n}$, then $\mathcal{C} A={\mathbf{c} A \mid \mathbf{c} \in \mathcal{C}} .$ Let $\mathcal{C}{1}$ and $\mathcal{C}{2}$ be linear codes over $\mathbb{F}{q}$ of length $n$. $\mathcal{C}{1}$ is permutation equivalent to $\mathcal{C}{2}$ provided $\mathcal{C}{2}=\mathcal{C}{1} P$ for some permutation matrix $P \in \mathbb{F}{q}^{n \times n} \cdot \mathcal{C}{1}$ is monomially equivalent to $\mathcal{C}{2}$ provided $\mathcal{C}{2}=\mathcal{C}{1} M$ for some monomial $\operatorname{matrix} M \in \mathbb{F}_{q}^{n \times n}$.Remark 1.8.2 Applying a permutation matrix to a code simply permutes the coordinates; applying a monomial matrix permutes and re-scales coordinates. Applying either a permutation or monomial matrix to a vector does not change its weight. Also applying either a permutation or monomial matrix to two vectors does not change the distance between these two vectors. There is a third more general concept of equivalence, involving semi-linear transformations, where two linear codes $\mathcal{C}{1}$ and $\mathcal{C}{2}$ over $\mathbb{F}{q}$ are equivalent provided one can be obtained from the other by permuting and re-scaling coordinates and then applying an automorphism of the field $\mathbb{F}{q}$. Note that applying such maps to a vector or to a pair of vectors preserves the weight of the vector and the distance between the two vectors, respectively; see [1008, Section 1.7] for further discussion of this type of equivalence. There are other concepts of equivalence that arise when the code may not be linear but has some specific algebraic structure (e.g., additive codes over $\mathbb{F}_{q}$ that are closed under vector addition but not necessarily closed under scalar multiplication). The common theme when defining equivalence of such codes is to use a set of maps which preserve distance between the two vectors, which preserve the algebraic structure under consideration, and which form a group under composition of these maps. We will follow this theme when we define equivalence of unrestricted codes at the end of this section.

## 数学代写|编码理论代写Coding theory代考|Bounds on Codes

In this section we present seven bounds relating the length, dimension or number of codewords, and minimum distance of an unrestricted code. The first five are considered upper bounds on the code size given length, minimum distance, and field size. By this, we mean that there does not exist a code of size bigger than the upper bound with the specified length, minimum distance, and field size. The last two are lower bounds on the size of a linear code. This means that a linear code can be constructed with the given length and minimum distance over the specified field having size equalling or exceeding the lower bound. We also give asymptotic versions of these bounds. Some of these bounds will be described using $A_{q}(n, d)$ and $B_{q}(n, d)$, which we now define.

Definition 1.9.1 For positive integers $n$ and $d, A_{q}(n, d)$ is the largest number of codewords in an $(n, M, d){q}$ code, linear or nonlinear. $B{q}(n, d)$ is the largest number of codewords in a $[n, k, d]{q}$ linear code. An $(n, M, d){q}$ code is optimal provided $M=A_{q}(n, d)$; an $[n, k, d]{q}$ linear code is optimal if $q^{k}=B{q}(n, d)$. The concept of ‘optimal’ can also be used in other contexts. Given $n$ and $d, k_{q}(n, d)$ denotes the largest dimension of a linear code over $\mathbb{F}{q}$ of length $n$ and minimum weight $d$; an $\left[n, k{q}(n, d), d\right]{q}$ code could be called ‘optimal in dimension’. Notice that $k{q}(n, d)=\log {q} B{q}(n, d)$. Similarly, $d_{q}(n, k)$ denotes the largest minimum distance of a linear code over $\mathbb{F}{q}$ of length $n$ and dimension $k$; an $\left[n, k, d{q}(n, k)\right]{q}$ may be called ‘optimal in distance’. Analogously, $n{q}(k, d)$ denotes the smallest length of a linear code over $\mathbb{F}{q}$ of dimension $k$ and minimum weight $d$; an $\left[n{q}(k, d), k, d\right]_{q}$ code might be called ‘optimal in length’.

Clearly $B_{q}(n, d) \leq A_{q}(n, d)$. On-line tables relating parameters of various types of codes are maintained by M. Grassl [845].

The following basic properties of $A_{q}(n, d)$ and $B_{q}(n, d)$ are easily derived; see [1008, Chapter 2.1].

## 数学代写|编码理论代写Coding theory代考|Puncturing, Extending, and Shortening Codes

(b) 定义\widehat{\mathcal{C}}=\left{c{1} c_{2} \cdots c_{n+1} \in \mathbb{F}{q}^{n+1} \mid c{1 } c_{2} \cdots c_{n} \in \mathcal{C}\right.$其中$\left.\sum_{i=1}^{n+1} c_{i}=0\right}\widehat{\mathcal{C}}=\left{c{1} c_{2} \cdots c_{n+1} \in \mathbb{F}{q}^{n+1} \mid c{1 } c_{2} \cdots c_{n} \in \mathcal{C}\right.$其中$\left.\sum_{i=1}^{n+1} c_{i}=0\right}，称为扩展代码。这是个[n+1,ķ,d^]q代码在哪里d^=d或者d+1. 发电机

\hat{H}=\left[\begin{数组}{ccc|c} 1 & \cdots & 1 & 1 \ \hline & & 0 \ & H & & \vdots \ & & & 0 \end{数组} \正确的]\hat{H}=\left[\begin{数组}{ccc|c} 1 & \cdots & 1 & 1 \ \hline & & 0 \ & H & & \vdots \ & & & 0 \end{数组} \正确的]
(c) 让小号是任何一组s坐标。让C(小号)是所有的代码字C是零小号. 穿刺C(小号)上小号结果是[n−s,ķ小号,d小号]q缩短的代码C小号在哪里d小号≥d. 如果C⊥有最小重量d⊥和s<d⊥， 然后ķ小号=ķ−s.

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## MATLAB代写

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