### 数学代写|编码理论代写Coding theory代考|ELEC7604

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## 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|MDS Codes

Maximum distance separable (MDS) codes are not only of theoretical interest, but rather important families of codes are of this type, such as Reed-Solomon codes (Section 1.14). An entire chapter is devoted to MDS codes in the book by MacWilliams and Sloane [1323. Chap. 11].

MDS codes are closely connected to many other structures in combinatorics and geometry. For example, an $[n, k, n-k+1]_{q}$ MDS code with dimension $k \geq 3$ corresponds to an $n$-arc in the projective geometry $\mathrm{PG}(k-1, q)$; see Chapter 14. Finite geometry is indeed a commonly used framework for studying MDS codes. In combinatorics, MDS codes correspond to certain orthogonal arrays.

Definition 3.3.18 An orthogonal array of size $N$, with $m$ constraints, $s$ levels, and strength $t$, denoted $\mathrm{OA}(N, m, s, t)$, is an $m \times N$ matrix with entries from $\mathbb{F}_{s}$, having the property that in every $t \times N$ submatrix, every $t \times 1$ column vector appears $\lambda=N / s^{t}$ (called the index) times.

Theorem 3.3.19 An $n \times q^{k}$ matrix with columns formed by the codewords of a linear $[n, k, n-k+1]{q} M D S$ code or an unvestricted $\left(n, q^{k}, n-k+1\right){q} M D S$ code is an $\mathrm{OA}\left(q^{k}, n, q, k\right)$, which has index $\lambda=1$.

Remark 3.3.20 As the codewords of an MDS code with dimension $k$ form an orthogonal array with strength $k$ and index 1 , such codes are systematic and any $k$ coordinates can be used for the message symbols.

In a paper [319] published by Bush in 1952 , the framework of orthogonal arrays is used to construct objects that we now know as Reed-Solomon codes. In that study it is also shown that for linear codes over $\mathbb{F}{q}$ with $k>q, n \leq k+1$ is a necessary condition for an $[n, k, n-k+1]{q}$ MDS code to exist, and that there are $[k+1, k, 2]{q}$ MDS codes. Such codes, and generally codes with parameters $[n, 1, n]{q},[n, n-1,2]{q}$, and $[n, n, 1]{q}$, are called trivial MDS codes.

For $k \leq q$, on the other hand, the following MDS Conjecture related to a question by Segre $[1638]$ in 1955 is still open.

Conjecture 3.3.21 (MDS) If $k \leq q$, then a linear $[n, k, n-k+1]_{q}$ MDS code exists exactly when $n \leq q+1$ unless $q=2^{h}$ and $k=3$ or $k=q-1$, in which case it exists exactly when $n \leq q+2$.

Remark 3.3.22 MDS codes are typically discussed in the linear case, but the parameters of the codes in Conjecture $3.3 .21$ conjecturally also cover the parameters for which unrestricted MDS codes exist.

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

The Hamming weight enumerator is defined in Definition $1.15 .1$ in Chapter 1. Recall that
$$\operatorname{Hwe}(x, y)=\sum_{i=0}^{n} A_{i}(\mathcal{C}) x^{i} y^{n-i}$$
Definition 4.2.1 A linear code $\mathcal{C}$ is called formally self-dual if $\mathcal{C}$ and its dual code $\mathcal{C}^{\perp}$ have the same weight enumerator, $\operatorname{Hwe} \mathcal{C}(x, y)=\operatorname{Hwe}_{\mathcal{C}}(x, y)$. A linear code is isodual if it is equivalent to its dual code.

Remark 4.2.2 Any isodual code is also formally self-dual, but there are formally self-dual codes that are neither isodual nor self-dual. The smallest length for which a formally selfdual code is not isodual is 14 , and there are 28 such codes amongst 6 weight enumerators [867]. Any self-dual code is also isodual and formally self-dual.
Example 4.2.3 The $[6,3,3]$ binary code $\mathcal{C}$ with a generator matrix
$$\left[\begin{array}{ll} 100 & 111 \ 010 & 110 \ 001 & 101 \end{array}\right]$$
is isodual. Its weight enumerator is $\operatorname{Hwe}_{\mathcal{C}}(x, y)=y^{6}+4 x^{3} y^{3}+3 x^{4} y^{2}$, and its automorphism group has order 24. Obviously, this code is not self-dual as it contains codewords with odd weight.

## 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].

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

MDS 代码与组合学和几何学中的许多其他结构密切相关。例如，一个[n,ķ,n−ķ+1]q带尺寸的 MDS 代码ķ≥3对应一个n- 射影几何中的弧磷G(ķ−1,q); 见第 14 章。有限几何确实是研究 MDS 代码的常用框架。在组合学中，MDS 码对应于某些正交数组。

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

[100111 010110 001101]

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## MATLAB代写

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