### 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|Equivalence and Isomorphism

The concepts of equivalence and isomorphism of codes are briefly discussed in Section 1.8. Generally, the term symmetry covers both of those concepts, especially when considering maps from a code onto itself, that is, automorphisms. Namely, such maps lead to groups under composition, and groups are essentially about symmetries. The group formed by all automorphisms of a code is, whenever the type of automorphisms is understood, simply called the automorphism group of the code. A subgroup of the automorphism group is called a group of automorphisms.

Symmetries play a central role when constructing as well as classifying codes: several types of constructions are essentially about prescribing symmetries, and one core part of classification is about dealing with maps and symmetries.

On a high level of abstraction, the same questions are asked for linear and unrestricted codes and analogous techniques are used. On a detailed level, however, there are significant differences between those two types of codes.

Consider codes of length $n$ over $\mathbb{F}{q}$. We have seen in Definition $1.8 .8$ that equivalence of unrestricted codes is about permuting coordinates and the elements of the alphabet, individually within each coordinate. All such maps form a group that is isomorphic to the wreath product $S{q} \backslash S_{n}$. For linear codes on the other hand, the concepts of permutation equivalence, monomial equivalence, and equivalence lead to maps that form groups isomorphic to $\mathrm{S}{n}, \mathbb{F}{q}^{} \backslash \mathrm{S}{n}$, and the semidirect product $\left(\mathbb{F}{q}^{} \imath \mathrm{S}{n}\right) \rtimes{\theta}$ Aut $\left(\mathbb{F}{q}\right)$, respectively, where $\mathbb{F}{q}^{}$ is the multiplicative group of $\mathbb{F}{q}$ and $\theta: \operatorname{Aut}\left(\mathbb{F}{q}\right) \rightarrow \operatorname{Aut}\left(\mathbb{F}{q}^{} \backslash \mathrm{S}{n}\right)$ is a group homomorphism.
Remark 3.2.1 For binary linear codes, all three types of equivalence coincide.

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

The obvious recurrent specific questions when studying equivalence (or isomorphism) of (linear and unrestricted) codes and the symmetries of such codes are the following:

1. Given two codes, $\mathcal{C}{1}$ and $\mathcal{C}{2}$, are these equivalent (isomorphic) or not?
2. Given a code $\mathcal{C}$, what is the automorphism group of $\mathcal{C}$ ?
The two questions are closely related, since if we are able to find all possible maps between two codes, $\mathcal{C}{1}$ and $\mathcal{C}{2}$, we can answer both of them (the latter by letting $\mathcal{C}{1}=\mathcal{C}{2}=$ C).
Invariants can be very useful in studying the first question.
Definition 3.2.15 An invariant is a property of a code that depends only on the abstract structure, that is, two equivalent (isomorphic) codes necessarily have the same value of an invariant.

Remark 3.2.16 Two inequivalent (non-isomorphic) codes may or may not have the same value of an invariant.

Example 3.2.17 The distance distribution is an invariant of codes. This invariant can be used to show that the two unrestricted $(4,3,2){3}$ codes $\mathcal{C}{1}={0000,0120,2121}$ and $\mathcal{C}{2}={1000,1111,2112}$ are inequivalent. Actually, to distinguish these codes an even less sensitive invariant suffices: the number of pairs of codewords with mutual Hamming distance 3 is 0 for $\mathcal{C}{1}$ and 1 for $\mathcal{C}_{2}$.

Since invariants are only occasionally able to provide the right answer to the first question above, alternative techniques are needed for providing the answer in all possible situations. One such technique relies on producing canonical representatives of codes.

Definition 3.2.18 Let $S$ be a set of possible codes, and let $r: S \rightarrow S$ be a map with the properties that (i) $r\left(\mathcal{C}{1}\right)=r\left(\mathcal{C}{2}\right)$ if and only if $\mathcal{C}{1}$ and $\mathcal{C}{2}$ are equivalent (isomorphic) and (ii) $r(\mathcal{C})=r(r(\mathcal{C})$ ). The canonical representative (or canonical form) of a code $\mathcal{C} \in S$ with respect to this map is $r(\mathcal{C})$.

To test whether two codes are equivalent (isomorphic), it suffices to test their canonical representatives for equality.

Remark 3.2.19 The number of equivalence (isomorphism) classes that can be handled when comparing canonical representatives is limited by the amount of computer memory available. However, in the context of classifying codes, there are actually methods that do not require any comparison between codes; see [1090].

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

Perfect codes were considered in the very first scientific papers in coding theory. We have already seen two types of perfect codes in Sections $1.10$ and 1.13. Hamming codes [895] have parameters
$$\left[n=\left(q^{m}-1\right) /(q-1), n-m, 3\right]{q}$$ and exist for $m \geq 2$ and prime powers $q$. Golay codes [820] have parameters $$[23,12,7]{2} \text { and }[11,6,5]{3} \text {. }$$ There are also some families of trivial perfect codes: codes containing one word, codes containing all codewords in the space, and $(n, 2, n){2}$ codes for odd $n$. If the order of the alphabet $q$ is a prime power, these are in fact the only sets of parameters for which (linear and unrestricted) perfect codes exist [1805, 1949].

Theorem 3.3.1 The nontrivial perfect linear codes over $\mathbb{F}{q}$, where $q$ is a prime power, are precisely the Hamming codes with parameters (3.1) and the Golay codes with parameters $(3.2)$. A nontrivial perfect unrestricted code (over $\mathbb{F}{q}, q$ a prime power) that is not equivalent to a linear code has the same length, size, and minimum distance as a Hamming code (3.1).
Although the remarkable Theorem 3.3.1 gives us a rather solid understanding of perfect codes, there are still many open problems in this area, including the following (a code with different alphabet sizes for different coordinates is called mixed):

Research Problem 3.3.2 Solve the existence problem for perfect codes when the size of the alphabet is not a prime power.
Research Problem 3.3.3 Solve the existence problem for perfect mixed codes.
Research Problem 3.3.4 Classify perfect codes, especially for the parameters covered by Theorem 3.3.1.

Since Theorem 3.3.1 covers alphabet sizes that are prime powers, that is, exactly the sizes for which finite fields and linear codes exist, Research Problems $3.3 .2$ to $3.3 .4$ are essentially about unrestricted codes (although many codes studied for Research Problem $3.3 .3$ have clear algebraic structures and close connections to linear codes).

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

1. 给定两个代码，C1和C2，这些是等价的（同构的）吗？
2. 给定一个代码C, 什么是自同构群C?
这两个问题密切相关，因为如果我们能够找到两个代码之间的所有可能映射，C1和C2，我们可以回答他们两个（后者通过让C1=C2=C）。
不变量在研究第一个问题时非常有用。
定义 3.2.15 不变量是仅依赖于抽象结构的代码的属性，即两个等价（同构）代码必然具有相同的不变量值。

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

[n=(q米−1)/(q−1),n−米,3]q并且存在米≥2和主要权力q. Golay 码 [820] 有参数

[23,12,7]2 和 [11,6,5]3. 还有一些平凡完美码族：包含一个词的码，包含空间中所有码字的码，以及(n,2,n)2奇数的代码n. 如果按字母顺序q是一个主要的力量，这些实际上是唯一存在（线性和无限制）完美代码的参数集 [1805, 1949]。

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## MATLAB代写

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