计算机代写|编码理论代写Coding theory代考|CS294-226

计算机代写|编码理论代写Coding theory代考|Weight Distributions

Thẻ weight distribution of a linear code determines thè weight distribution of its dual code via a series of equations, called the MacWilliams Identities or the MacWilliams Equations. They were first developed by F. J. MacWilliams in [1319]. There are in fact several equivalent formulations of these equations. Among these are the Pless Power Moments discovered by V. S. Pless [1513]. The most compact form of these identities is expressed in a single polynomial equation relating the weight distribution of a code and its dual.

Definition 1.15.1 Let $\mathcal{C}$ be a linear code of length $n$ over $\mathbb{F}q$ with weight distribution $A_i(\mathcal{C})$ for $0 \leq i \leq n$. Let $x$ and $y$ be independent indeterminates over $\mathbb{F}_q$. The (Hamming) weight enumerator of $\mathcal{C}$ is defined to be $$\operatorname{Hwe}{\mathcal{C}}(x, y)=\sum_{i=0}^n A_i(\mathcal{C}) x^i y^{n-i} .$$
The formulation of the Pless Power Moments involves Stirling numbers.
Definition 1.15.2 The Stirling numbers $S(r, \nu)$ of the second kind are defined for nonnegative integers $r, \nu$ by the equation
$$S(r, \nu)=\frac{1}{\nu !} \sum_{i=0}^\nu(-1)^{\nu-i}\left(\begin{array}{l} \nu \ i \end{array}\right) i^r ;$$
$\nu ! S(r, \nu)$ is the number of ways to distribute $r$ distinct objects into $\nu$ distinct boxes with no box left empty.

计算机代写|编码理论代写Coding theory代考|Encoding

Figure $1.1$ shows a simple communication channel that includes a component called an encoder, in which a message is encoded to produce a codeword. In this section we examine two encoding processes.

As in Figure 1.1, a message is any of the $q^k$ possible $k$-tuples $\mathbf{x} \in \mathbb{F}_q^k$. The encoder will convert $\mathbf{x}$ to an $n$-tuple $\mathbf{c}$ from a code $\mathcal{C}$ over $\mathbb{F}_q$ with $q^k$ codewords; that codeword will then be transmitted over the communication channel.

Suppose that $\mathcal{C}$ is an $[n, k, d]_q$ linear code with generator matrix $G$ and parity check matrix $H$. We first describe an encoder that uses the generator matrix $G$. The most common way to encode the message $\mathbf{x}$ is as $\mathbf{x} \mapsto \mathbf{c}=\mathbf{x} G$. If $G$ is replaced by another generator matrix, the encoding of $\mathrm{x}$ will, of course, be different. A nice relationship exists between message and codeword if $G$ is in standard form $\left[I_k \mid A\right]$. In that case the first $k$ coordinates of the codeword $\mathbf{c}$ are the information symbols $\mathbf{x}$ in order; the remaining $n-k$ symbols are the parity check symbols, that is, the redundancy added to $x$ in order to help recover $x$ if errors occur during transmission. A similar relationship between message and codeword can exist even if $G$ is not in standard form. Specifically, suppose there exist column indices $i_1, i_2, \ldots, i_k$ such that the $k \times k$ matrix consisting of these $k$ columns of $G$ is the $k \times k$ identity matrix. In that case the message is found in the $k$ coordinates $i_1, i_2, \ldots, i_k$ of the codeword scrambled but otherwise unchanged; that is, the message symbol $x_j$ is in component $i_j$ of the codeword. If this occurs where the message is embedded in the codeword, we say that the encoder is a systematic encoder of $\mathcal{C}$. We can always force an encoder to be systematic. For example, if $G$ is row reduced to a matrix $G_1$ in reduced row echelon form, $G_1$ remains a generator matrix of $\mathcal{C}$ by Remark $1.4 .3$; the encoding $\mathbf{x} \mapsto \mathbf{c}=\mathbf{x} G_1$ is systematic as $G_1$ has $k$ columns which together form $I_k$. Another way to force an encoder to be systematic is as follows. By Theorem 1.8.4, $\mathcal{C}$ is permutation equivalent to an $[n, k, d]_q$ code $\mathcal{C}^{\prime}$ with generator matrix $G^{\prime}$ in standard form. If the code $\mathcal{C}^{\prime}$ is used in place of $\mathcal{C}$, the encoder $\mathrm{x} \mapsto \mathrm{x} G^{\prime}$ is a systematic encoder of $\mathcal{C}^{\prime}$.

编码理论代考

计算机代写|编码理论代写Coding theory代考|Weight Distributions

Pless Power Moments 的公式涉及斯特林数。

$$S(r, \nu)=\frac{1}{\nu !} \sum_{i=0}^\nu(-1)^{\nu-i}(\nu i) i^r$$
$\nu ! S(r, \nu)$ 是分配方式的数量 $r$ 不同的物体成 $\nu$ 不同的盒子，没有盒子空着。

计算机代写|编码理论代写Coding theory代考|Encoding

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