### 数学代写|编码理论代写Coding theory代考|elec 3004

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## 数学代写|编码理论代写Coding theory代考|The Minimum Distances of BCH Codes

It follows from Theorem 2.4.1 that a cyclic code with designed distance $\delta$ has minimum weight at least $\delta$. It is possible that the actual minimum distance is equal to the designed distance. Sometimes the actual minimum distance is much larger than the designed distance.
A codeword $\left(c_{0}, \ldots, c_{n-1}\right)$ of a linear code $\mathcal{C}$ is even-like if $\sum_{j=0}^{n-1} c_{j}=0$, and oddlike otherwise. The weight of an even-like (respectively odd-like) codeword is called an even-like weight (respectively odd-like weight). Let $\mathcal{C}$ be a primitive narrow-sense BCH code of length $n=q^{m}-1$ over $\mathbb{F}{q}$ with designed distance $\delta$. The defining set is then $T(1, \delta)=C{1} \cup C_{2} \cup \cdots \cup C_{\delta-1}$. The following theorem provides useful information on the minimum weight of narrow-sense primitive BCH codes.

Theorem 2.6.4 Let $\mathcal{C}$ be the narrow-sense primitive $B C H$ code of length $n=q^{m}-1$ over $\mathbb{F}{q}$ with designed distance $\delta$. Then the minimum weight of $\mathcal{C}$ is its minimum odd-like weight. The coordinates of the narrow-sense primitive BCH code $\mathcal{C}$ of length $n=q^{m}-1$ over $\mathbb{F}{q}$ with designed distance $\delta$ can be indexed by the elements of $\mathbb{F}{q^{m}}^{}$, and the extended coordinate in the extended code $\widehat{\mathcal{C}}$ can be indexed by the zero element of $\mathbb{F}{q^{m}}$. The general affine group $\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)$ then acts on $\mathbb{F}{q^{m}}$ and also on $\widehat{\mathcal{C}}$ doubly transitively, where $$\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m}}^{}, b \in \mathbb{F}{q^{m}}\right} .$$ Since $\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)$ is transitive on $\mathbb{F}{q^{m}}$, it is a subgroup of the permutation automorphism group of $\widehat{\mathcal{C}}$. Theorem 2.6.4 then follows.

In the following cases, the minimum distance of the BCH code $\mathcal{C}_{(q, n, \delta, b)}$ is known. We first have the following [1323, p. 260].

## 数学代写|编码理论代写Coding theory代考|The Dimensions of BCH Codes

The dimension of the BCH code $\mathcal{C}_{(q, n, \delta, b)}$ with defining set $T(b, \delta)$ in $(2.2)$ is $n-|T(b, \delta)|$. Since $|T(b, \delta)|$ may have a very complicated relation with $n, q, b$ and $\delta$, the dimension of the BCH code cannot be given exactly in terms of these parameters. The best one can do in general is to develop tight lower bounds on the dimension of BCH codes. The next theorem introduces such bounds [1008, Theorem 5.1.7].

Theorem 2.6.8 Let $\mathcal{C}$ be an $[n, \kappa] B C H$ code over $\mathbb{F}{q}$ of designed distance $\delta$. Then the following statements hold. (a) $\kappa \geq n-\operatorname{ord}{n}(q)(\delta-1)$.
(b) If $q=2$ and $\mathcal{C}$ is a narrow-sense $B C H$ code, then $\delta$ can be assumed odd; furthermore if $\delta=2 w+1$, then $\kappa \geq n-\operatorname{ord}_{n}(q) w$.

The bounds in Theorem $2.6 .8$ may not be improved for the general case, as demonstrated by the following example. However, in some special cases, they could be improved.

Example 2.6.9 Note that $m=\operatorname{ord}{15}(2)=4$, and the 2-cyclotomic cosets modulo 15 are \begin{aligned} &C{0}={0}, C_{1}={1,2,4,8}, C_{3}={3,6,9,12}, \ &C_{5}={5,10}, C_{7}={7,11,13,14} . \end{aligned}
Let $\gamma$ be a generator of $\mathbb{F}_{2^{4}}^{*}$ with $\gamma^{4}+\gamma+1=0$ and let $\alpha=\gamma^{\left(2^{4}-1\right) / 15}=\gamma$ be the primitive $15^{\text {th }}$ root of unity.

When $(b, \delta)=(0,3)$, the defining set $T(b, \delta)={0,1,2,4,8}$, and the binary cyclic code has parameters $[15,10,4]$ and generator polynomial $x^{5}+x^{4}+x^{2}+1$. In this case, the actual minimum weight is more than the designed distance, and the dimension is larger than the bound in Theorem 2.6.8(a).

When $(b, \delta)=(1,3)$, the defining set $T(b, \delta)={1,2,4,8}$, and the binary cyclic code has parameters $[15,11,3]$ and generator polynomial $x^{4}+x+1$. It is a narrow-sense BCH

code. In this case, the actual minimum weight is equal to the designed distance, and the dimension reaches the bound in Theorem $2.6 .8(\mathrm{~b})$.

When $(b, \delta)=(2,3)$, the defining set $T(b, \delta)={1,2,3,4,6,8,9,12}$, and the binary cyclic code has parameters $[15,7,5]$ and generator polynomial $x^{8}+x^{7}+x^{6}+x^{4}+1$. In this case, the actual minimum weight is more than the designed distance, and the dimension achieves the bound in Theorem 2.6.8(a).

When $(b, \delta)=(1,5)$, the defining set $T(b, \delta)={1,2,3,4,6,8,9,12}$, and the binary cyclic code has parameters $[15,7,5]$ and generator polynomial $x^{8}+x^{7}+x^{6}+x^{4}+1$. In this case, the actual minimum weight is equal to the designed distance, and the dimension is larger than the bound in Theorem 2.6.8(a). Note that the three pairs $\left(b_{1}, \delta_{1}\right)=(2,3),\left(b_{2}, \delta_{2}\right)=$ $(2,4)$ and $\left(b_{3}, \delta_{3}\right)=(1,5)$ define the same binary cyclic code with generator polynomial $x^{8}+x^{7}+x^{6}+x^{4}+1$. Hence the maximum designed distance of this $[15,7,5]$ cyclic code is $5 .$

When $(b, \delta)=(3,4)$, the defining set $T(b, \delta)={1,2,3,4,5,6,8,9,10,12}$, and the binary cyclic code has parameters $[15,5,7]$ and generator polynomial $x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1$. In this case, the actual minimum weight is more than the designed distance, and dimension is larger than the bound in Theorem 2.6.8(a).
The following is a general result on the dimension of BCH codes [47].

Duadic codes are a family of cyclic codes and are generalizations of the quadratic residue codes. Binary duadic codes were defined in [1220] and were generalized to arbitrary finite fields in $[1517,1519]$. Some duadic codes have very good parameters, while some have very bad parameters. The objective of this section is to give a brief introduction of duadic codes.
As before, let $n$ be a positive integer and $q$ a prime power with $g c d(n, q)=1$. Let $S_{1}$ and $S_{2}$ be two subsets of $Z_{n}$ such that

• $S_{1} \cap S_{2}=\emptyset$ and $S_{1} \cup S_{2}=\mathbb{Z}_{n} \backslash{0}$, and
• both $S_{1}$ and $S_{2}$ are a union of some $q$-cyclotomic cosets modulo $n$.
If there is a unit $\mu \in \mathbb{Z}{n}$ such that $S{1} \mu=S_{2}$ and $S_{2} \mu=S_{1}$, then $\left(S_{1}, S_{2}, \mu\right)$ is called a splitting of $\mathbb{Z}_{n}$.

Recall that $m:=\operatorname{ord}{n}(q)$ and $\alpha$ is a primitive $n^{\text {th }}$ root of unity in $\mathbb{F}{q^{m}}$. Let $\left(S_{1}, S_{2}, \mu\right)$ be a splitting of $\mathbb{Z}{m}$. Define $$g{i}(x)=\prod_{i \in S_{i}}\left(x-\alpha^{i}\right) \text { and } \tilde{g}{i}(x)=(x-1) g{i}(x)$$
for $i \in{1,2}$. Since both $S_{1}$ and $S_{2}$ are unions of $q$-cyclotomic cosets modulo $n$, both $g_{1}(x)$ and $g_{2}(x)$ are polynomials over $\mathbb{F}{q}$. The pair of cyclic codes $\mathcal{C}{1}$ and $\mathcal{C}{2}$ of length $n$ over $\mathbb{F}{q}$ with generator polynomials $g_{\widetilde{r}}(x)$ and $g_{2}(x)$ are called odd-like duadic codes, and the pair of cyclic codes $\widetilde{\mathcal{C}}{1}$ and $\widetilde{\mathcal{C}}{2}$ of length $n$ over $\mathbb{F}{q}$ with generator polynomials $\widetilde{g}{1}(x)$ and $\tilde{g}_{2}(x)$ are called even-like duadic codes.

By definition, $\mathcal{C}{1}$ and $\mathcal{C}{2}$ have parameters $[n,(n+1) / 2]$ and $\widetilde{\mathcal{C}}{1}$ and $\tilde{\mathcal{C}}{2}$ have parameters $[n,(n-1) / 2]$. For odd-like duadic codes, we have the following result [1008, Theorem 6.5.2].
Theorem 2.7.1 (Square Root Bound) Let $\mathcal{C}{1}$ and $\mathcal{C}{2}$ be a pair of odd-like duadic codes of length $n$ over $\mathbb{F}{q}$. Let $d{o}$ be their (common) minimum odd-like weight. Then the following hold.
(a) $d_{o}^{2} \geq n$.
(b) If the splitting defining the duadic codes is given by $\mu=-1$, then $d_{o}^{2}-d_{o}+1 \geq n$.
(c) Suppose $d_{o}^{2}-d_{o}+1=n$, where $d_{o}>2$, and assume that the splitting defining the duadic codes is given by $\mu=-1$. Then $d_{o}$ is the minimum weight of both $\mathcal{C}{1}$ and $\mathcal{C}{2}$.

## 数学代写|编码理论代写Coding theory代考|The Minimum Distances of BCH Codes

\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m} }^{}, b \in \mathbb{F}{q^{m}}\right} 。\mathrm{GA}{1}\left(\mathbb{F}{q^{m}}\right)=\left{a x+b \mid a \in \mathbb{F}{q^{m} }^{}, b \in \mathbb{F}{q^{m}}\right} 。自从G一个1(Fq米)是可传递的Fq米，它是置换自同构群的一个子群C^. 定理 2.6.4 随之而来。

## 数学代写|编码理论代写Coding theory代考|The Dimensions of BCH Codes

BCH码的维度C(q,n,d,b)带有定义集吨(b,d)在(2.2)是n−|吨(b,d)|. 自从|吨(b,d)|可能有很复杂的关系n,q,b和d，BCH码的维数不能根据这些参数准确给出。一般来说，最好的办法是在 BCH 代码的维度上制定严格的下限。下一个定理引入了这样的界限[1008，定理 5.1.7]。

(b) 如果q=2和C是狭义的乙CH代码，然后d可以假设为奇数；此外，如果d=2在+1， 然后ķ≥n−单词n⁡(q)在.

C0=0,C1=1,2,4,8,C3=3,6,9,12, C5=5,10,C7=7,11,13,14.

• 小号1∩小号2=∅和小号1∪小号2=从n∖0， 和
• 两个都小号1和小号2是一些人的联合q-分圆陪集模n.
如果有单位μ∈从n这样小号1μ=小号2和小号2μ=小号1， 然后(小号1,小号2,μ)被称为分裂从n.

G一世(X)=∏一世∈小号一世(X−一个一世) 和 G~一世(X)=(X−1)G一世(X)

（一个）d○2≥n.
(b) 如果定义二元码的分裂由下式给出μ=−1， 然后d○2−d○+1≥n.
(c) 假设d○2−d○+1=n， 在哪里d○>2，并假设定义二元码的分裂由下式给出μ=−1. 然后d○是两者的最小权重C1和C2.

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