计算机代写|编码理论代写Coding theory代考|ELEN90030

计算机代写|编码理论代写Coding theory代考|Subfield Subcodes

Let $\mathcal{C}$ be an $[n, \kappa]{q^t}$ code. The subfield subcode $\left.\mathcal{C}\right|{\mathbb{F}q}$ of $\mathcal{C}$ with respect to $\mathbb{F}_q$ is the set of codewords in $\mathcal{C}$ each of whose components is in $\mathbb{F}_q$. Since $\mathcal{C}$ is linear over $\mathbb{F}{q^t},\left.\mathcal{C}\right|_{\mathbb{F}_q}$ is a linear code over $\mathbb{F}_q$.

The dimension, denoted $\kappa_q$, of the subfield subcode $\left.\mathcal{C}\right|_{F_q}$ may not have an elementary relation with that of the code $\mathcal{C}$. However, we have the following lower and upper bounds on $\kappa_q$.

Theorem 2.2.1 Let $\mathcal{C}$ be an $[n, \kappa]{q^t}$ code. Then $\left.\mathcal{C}\right|{\mathbb{F}q}$ is an $\left[n, \kappa_q\right]$ code over $\mathbb{F}_q$, where $\kappa \geq \kappa_q \geq n-t(n-\kappa)$. If $\mathcal{C}$ has a basis of codewords in $\mathbb{F}_q^n$, then this is also a basis of $\mathcal{C}{\mathbb{F}q}$ and $\left.\mathcal{C}\right|{\mathbb{F}_q}$ has dimension $\kappa$.

Example 2.2.2 The Hamming code $\mathcal{H}{3,2^2}$ over $\mathbb{F}{2^2}$ has parameters $[21,18,3]4$. The subfield subcode $\left.\mathcal{H}{3,2^2}\right|_{\mathrm{F} ;}$ is a $[21,16,3]_2$ code with parity check matrix
$$\left[\begin{array}{lllllllllllllllllllll} 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 \ 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \end{array}\right] .$$
In this case, $n=21, \kappa=18$, and $n-t(n-\kappa)=15$. Hence $\kappa_q=16$, which is very close to $n-t(n-\kappa)=15$.

The following is called Delsarte’s Theorem, which exhibits a dual relation between subfield subcodes and trace codes. This theorem is very useful in the design and analysis of linear codes.
Theorem 2.2.3 (Delsarte) Let $\mathcal{C}$ be a linear code of length $n$ over $\mathbb{F}{q^t}$. Then $$\left(\left.\mathcal{C}\right|{\mathbb{F}q}\right)^{\perp}=\operatorname{Tr}{q^t / q}\left(\mathcal{C}^{\perp}\right),$$
where $\operatorname{Tr}{q^t / q}\left(\mathcal{C}^{\perp}\right)=\left{\left(\operatorname{Tr}{q^t / q}\left(v_1\right), \ldots, \operatorname{Tr}_{q^t / q}\left(v_n\right)\right) \mid\left(v_1, \ldots, v_n\right) \in \mathcal{C}^{\perp}\right}$.
Theorems $2.2 .1$ and $2.2 .3$ work for all linear codes, including cyclic codes. Their proofs could be found in $[1008$, Section 3.8]. We shall need them later.

计算机代写|编码理论代写Coding theory代考|Fundamental Constructions of Cyclic Codes

In Section 1.12, it was shown that every cyclic code of length $n$ over $\mathbb{F}_q$ can be generated by a generator polynomial $g(x) \in \mathbb{F}_q[x]$. The objective of this section is to describe several other fundamental constructions of cyclic codes over finite fields. By a fundamental construction, we mean a construction method that can produce every cyclic code over any finite field.

An element $e$ in a commutative ring $\mathcal{R}$ is called an idempotent if $e^2=e$. The ring $\mathcal{R}_{(n, q)}$ has in general quite a number of idempotents. Besides its generator polynomial, many other polynomials can generate a cyclic code $\mathcal{C}$. Let $\mathcal{C}$ be a cyclic code over $\mathbb{F}_q$ with generator polynomial $g(x)$. It is easily seen that a polynomial $f(x) \in \mathbb{F}_q[x]$ generates $\mathcal{C}$ if and only if $\operatorname{gcd}\left(f(x), x^n-1\right)=g(x)$.

If an idempotent $e(x) \in \mathcal{R}_{(n, q)}$ generates a cyclic code $\mathcal{C}$, it is then unique in this ring and called the generating idempotent. Given the generator polynomial of a cyclic code, one can compute its generating idempotent with the following theorem [1008, Theorem 4.3.3].
Theorem 2.3.1 Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}_q$ with generator polynomial $g(x)$. Let $h(x)=\left(x^n-1\right) / g(x)$. Then $\operatorname{gcd}(g(x), h(x))=1$, as it was assumed that $\operatorname{gcd}(n, q)=1$. Employing the Extended Euclidean Algorithm, one computes two polynomials a $a(x) \in \mathbb{F}_q[x]$ and $b(x) \in \mathbb{F}_q[x]$ such that $1=a(x) g(x)+b(x) h(x)$. Then $e(x)=a(x) g(x) \bmod \left(x^n-1\right)$ is the generating idempotent of $\mathcal{C}$.

The polynomial $h(x)$ in Theorem $2.3 .1$ is called the parity check polynomial of $\mathcal{C}$. Given the generating idempotent of a cyclic code, one obtains the generator polynomial of this code as follows [1008, Theorem 4.3.3].

Theorem 2.3.2 Let $\mathcal{C}$ be a cyclic code over $\mathbb{F}_q$ with generating idempotent $(x)$. Then the generator polynomial of $\mathcal{C}$ is given by $g(x)=\operatorname{gcd}\left(e(x), x^n-1\right)$, which is computed in $\mathbb{F}_q[x]$.

编码理论代考

计算机代写|编码理论代写Coding theory代考|Subfield Subcodes

$\left[\begin{array}{llllllllllllllllllllllllllll}1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1\end{array}\right.$

$$(\mathcal{C} \mid \mathbb{F} q)^{\perp}=\operatorname{Tr} q^t / q\left(\mathcal{C}^{\perp}\right),$$

计算机代写|编码理论代写Coding theory代考|Fundamental Constructions of Cyclic Codes

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