### 数学代写|编码理论代写Coding theory代考|MTH3018

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• Foundations of Data Science 数据科学基础

## 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 $\mathrm{S}{q} \geq \mathrm{S}{n}$. For linear codes on the other hand, the concepts of permutation equivalence, monomial equivalence, and equivalence lead to maps that form groups isomorphic to $\mathrm{S}{n}, \mathbb{F}{q}^{}\left\langle\mathrm{~S}{n}\right.$, and the semidirect product $\left(\mathbb{F}{q}^{}\left\langle\mathrm{~S}{n}\right) \rtimes_{\theta}\right.$ 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.

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

A code of size $M$ is a subset of $M$ vectors from the $n$-dimensional vector space over $\mathbb{F}{q}$ which fulfills some requirements depending on the type of code. The number of ways to choose $M$ arbitrary vectors from such a space is $\left({ }^{q}\right)$, which becomes astronomically large already for rather small parameters. (This is obviously the total number of $(n, M){q}$ codes.) Although no general conclusion regarding the hardness of solving construction and classification problems can be drawn from this number, the number does give a clue that the limit of what is feasible might be reached quite early. Indeed, this is what happens, but perhaps not as early as one would think.

Example 3.2.2 In some special cases – in particular, for perfect codes quite large unrestricted codes have been classified, such as the $(23,4096,7){2}$ code (the binary Golay code is unique $[1732]$; see also $[525]$ ) and the $(15,2048,3){2}$ codes (with the parameters of a Hamming code; there are 5983 such codes [1472]).

But what can be done if we go beyond parameters for which the size of an optimal code can be determined and the optimal codes can be classified? Analytical upper bounds and constructive lower bounds on the size of codes can still be used. One way to speed up computer-aided constructive techniques-some of which are discussed in Chapter 23 -is to restrict the search by imposing a structure on the codes. This is a two-edged sword: the search space is reduced, but good codes might not have that particular structure. Hence some experience is of great help in tuning the search. A very common approach is that of prescribing symmetries (automorphisms).

Remark 3.2.3 In the discussion of groups in the context of automorphism groups of codes, we are not only interested in the abstract group but in the group and its action. This is implicitly understood in the sequel when talking about one particular group or all groups of certain orders. For example, “prescribing a group” means “prescribing a group and its action” and “considering all groups” means “considering all groups and all possible actions of those groups”.

By prescribing a group $G$, the $n$-dimensional vector space is partitioned into orbits of vectors. The construction problem then becomes a problem of finding a set of those orbits rather than finding a set of individual vectors. It must further be checked that the orbits themselves are feasible; an orbit whose codewords do not fulfill the minimum distance criterion can be discarded immediately.

Remark 3.2.4 An $[n, k]_{q}$ linear code can be viewed as an unrestricted code which contains the all-zero codeword and has a particular group of automorphisms $G$ of order $q^{k}$, which only permutes elements of the alphabet, individually within each coordinate.

## 数学代写|编码理论代写Coding theory代考|Some Central Classes of Codes

By Definition 1.9.1, the maximum size of error-correcting codes with length $n$ and minimum distance $d$ are given by the functions $A_{q}(n, d)$ and $B_{q}(n, d)$ for unrestricted and linear codes, respectively. Most general bounds on these functions, such as those in Section 1.9,

consider upper bounds and are about nonexistence of codes. Lower bounds, on the other hand, are typically obtained by constructing explicit codes. Especially for small parameters, many best known codes have been obtained on a case-by-case basis. One possible approach for finding such codes is that of prescribing symmetries as discussed in Section $3.2 .1-$ and carrying out a computer search; see Chapter $23 .$

In some rare situations, there exist codes that attain some general upper bounds. For such parameters, the problem of finding the size of an optimal code is then settled. When this occurs and the upper bound is the Sphere Packing Bound, we get perfect codes (Definition 1.9.8), and when the upper bound is the Singleton Bound, we get maximum distance separable (MDS) codes (Definition 1.9.12). In this section we will take a glance at these two types of codes as well as general binary linear and unrestricted codes.

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