Galois环上的加性循环码：枚举与构造方法

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"这篇文章主要探讨了Galois环上的加性循环码的枚举与构造。作者们在特征为p的Galois环R=GR(p^l, l)上研究了这种编码，并且l是素数。文章首先提出了加性循环码的规范形式分解，这一分解对于构建这类编码以及计算其数量至关重要。接着，利用R的规范形式分解和Z_p扩展Galois环上的线性码，他们推导出了R上每个加性循环码的跟踪对偶码。关键词包括：加性循环码、Galois环、线性码、对偶码、跟踪内积、自对偶码和准循环码。" 正文：在信息技术和通信领域，编码理论是一个核心研究方向，它涉及到数据传输的可靠性和效率。本文重点讨论了一种特殊的编码类型——加性循环码，它在Galois环上被定义和操作。Galois环是代数学中的一个重要概念，它是具有特定性质的环结构，特别是它们的乘法运算满足交换律，而且是域的局部化。在这种环上，加性循环码是那些在环的加法运算下封闭的码，且在环的乘法下满足循环性质。文章首先介绍了在R=GR(p^l, l)上加性循环码的规范形式分解。这种分解是一种有效的工具，可以帮助我们理解和构造这类编码。通过规范形式分解，我们可以将复杂的编码结构简化为基本单元，从而更容易地研究其性质和计数问题。了解这些编码的数量对于理解编码空间的大小和可能的编码方案的多样性至关重要。接下来，作者们转向了跟踪对偶码的概念。在Galois环上，跟踪内积是一种特殊的内积，它可以用来定义码的对偶。对偶码在码论中有着重要的地位，因为它可以提供错误检测和校正的能力。通过对R的规范形式分解以及Z_p扩展Galois环上的线性码的利用，他们能够找出每一个加性循环码的跟踪对偶码。这种方法揭示了编码之间的关系，有助于设计出更有效的编码系统。此外，文章还提及了自对偶码和准循环码。自对偶码是其自身对偶的码，它们在编码设计中有特殊的吸引力，因为它们提供了对称的编码结构，有利于提高编码效率。准循环码则是一类接近于循环码的编码，它们在码字的处理上具有优势，特别是在硬件实现时。这篇研究工作深入探讨了Galois环上的加性循环码，不仅给出了它们的构造方法，还研究了如何通过跟踪对偶码来增强编码性能。这些成果对于编码理论和实际的通信系统设计都有重要的理论价值和应用前景。通过这样的研究，我们可以开发出更高效、更可靠的编码技术，这对于未来的通信网络和数据存储系统的优化至关重要。

Y. Cao et al. / Discrete Mathematics 338 (2015) 922–937 925

(ii) R

= I

⊕ I

⊕ · · · ⊕ I

, where ε

(X) is the multiplicative identity of I

for each 0 ≤ i ≤ s. This direct sum is a ring

direct sum in that I

= {0} if i = j.

(iii) For any r + 1 ≤ i ≤ s, we have I

= I

i,0

⊕ I

i,1

⊕ · · · ⊕ I

i,l−1

, where ε

i,h

(X) is the multiplicative identity of I

i,h

for each

0 ≤ h ≤ l − 1. This direct sum is a ring direct sum in that I

i,h

i,j

= {0} if h = j.

(iv) For each 0 ≤ i ≤ r, the mapping ψ

: R[X ]/(m

(X)) → I

defined by

f (X) + (m

(X)) → ε

(X)f (X ) + (X

− 1) (∀f (X) ∈ R[X])

is a ring isomorphism. Hence I

is an extension Galois ring of R. In particular, I

is a free R-module of rank κ

(v) For each r + 1 ≤ i ≤ s and 0 ≤ h ≤ l − 1, the mapping ψ

i,h

: R[X ]/(m

i,h

(X)) → I

i,h

defined by

f (X) + (m

i,h

(X)) → ε

i,h

(X)f (X ) + (X

− 1) (∀f (X) ∈ R[X])

is a ring isomorphism. Hence I

i,h

is an extension Galois ring of R. In particular, I

i,h

is a free R-module of rank κ

i,h

From now on, for any a(X ) =



n−1

k=0

∈ R

with a

∈ R, we define

φ(a(X)) =

n−1



k=0

φ(a

, µ(a(X)) =

n−1



k=0

−k

= X





n−1



k=0

n−k

Then we have the following results.

• φ is a ring automorphism of R

of multiplicative order l.

• µ is a ring automorphism of R

of multiplicative order 2. For any a(X ) ∈ R

, we will denote



a(X) = µ(a(X )) in some

cases.

• φµ = µφ.

• R

(p)

is precisely the subring of R

whose elements are fixed by φ; i.e., R

(p)

is the set of polynomials in R

with all

coefficients in Z

In the rest of this paper, we denote

ϱ(Y ) = (Y − ω)(Y − ω

) · · · (Y − ω

l−1

Then ϱ(Y ) is a monic basic primitive polynomial in Z

[Y ] of degree l as ord(ω) = p

− 1, R = Z

[ω]

∼

[Y ]/(ϱ(Y )) and

every element c of R can be uniquely expressed as c

+ c

ω + · · · + c

l−1

where c

, c

, . . . , c

l−1

∈ Z

(cf. [15] Theo-

rem 14.8). Therefore, R is a free Z

-module of rank l and {1, ω, . . . , ω

l−1

} is a Z

-basis of the Galois ring R. In the following,

we gather some facts about the image of I

and I

i,h

under φ, and about the relationship between K

and I

Theorem 2.4. Using the notations above, the following hold.

(i) φ(ε

(X)) = ε

(X) for all 0 ≤ i ≤ s, and φ(ε

i,h

(X)) = ε

i,h+1(mod l)

(X) for all r + 1 ≤ i ≤ s and 0 ≤ h ≤ l − 1.

(ii) φ(I

) = I

for all 0 ≤ i ≤ s, and φ(I

i,h

) = I

i,h+1(mod l)

for all r + 1 ≤ i ≤ s and 0 ≤ h ≤ l − 1.

(iii) K

= I

∩ R

(p)

and hence I

is a K

-module for all 0 ≤ i ≤ s.

(iv) If 0 ≤ i ≤ r, then I

is a free K

-module of rank l with a basis given by {ε

(X), ωε

(X), . . . , ω

l−1

(X)}.

(v) If r + 1 ≤ i ≤ s, then I

is a free K

-module of rank l with a basis given by {ε

i,0

(X), ε

i,1

(X), . . . , ε

i,l−1

(X)}.

Proof. (i) Because ε

(X) ∈ Z

[X], φ(ε

(X)) = ε

(X). The remainder of (i) follows from the definitions of ε

i,h

(X) and φ.

(ii) It follows from (i) and the definitions of I

and I

i,h

(iii) Obviously, K

= ε

(X)R

(p)

⊆ I

∩ R

(p)

. Conversely, let a(X) ∈ I

∩ R

(p)

. Then a(X ) ∈ R

(p)

and a(X ) ∈ I

. Since

(X) is the multiplicative identity of I

by Lemma 2.3(ii), we have a(X ) = ε

(X)a(X ) ∈ K

(iv) Let 0 ≤ i ≤ r. Then gcd(κ

, l) = 1. By Lemma 2.2, K

is a Galois ring of characteristic p

and cardinality p

ϵκ

. Since

ω is a primitive (p

− 1)th root of unity, by ω ∈ R ⊆ R

and Lemmas 2.2 and 2.3 ε

(X)ω is a primitive (p

− 1)th root

of unity contained in I

. But K

is a subring of I

, and the minimal polynomial of ε

(X)ω over K

is g

(Y ) =



d−1

j=0

(Y −

(ε

(X)ω)

)

) where d = min{a ∈ Z

| (p

)

≡ 1 (mod p

− 1)} =

gcd(κ

,l)

= l. From this we deduce that K

[ε

(X)ω]

∼

[Y ]/(g

(Y )) which is a Galois ring of characteristic p

and cardinality |K

. Therefore, a K

-basis of K

[ε

(X)ω] is given

by {ε

(X), ε

(X)ω, . . . , ε

(X)ω

l−1

}. Obviously, we have K

[ε

(X)ω] ⊆ I

. By Lemma 2.3(iv) it follows that |I

| = p

ϵκ

[ε

(X)ω]|. Hence I

= K

[ε

(X)ω].

(v) Let r + 1 ≤ i ≤ s and denote S

= {



l−1

h=0

(α) | α ∈ I

i,0

}. First, we claim that K

= S

. In fact, for any α ∈ I

i,0

we have φ

(α) ∈ I

i,h

by (ii), which then implies



l−1

h=0

(α) ∈



l−1

h=0

i,h

= I

by Lemma 2.3(iii), and so S

⊆ I

. Since the

multiplicative order of φ is l, we have φ(



l−1

h=0

(α)) =



l−1

h=0

h+1

(α) =



l−1

h=0

(α), which implies



l−1

h=0

(α) ∈ R

(p)

and so



l−1

h=0

(α) ∈ I

∩ R

(p)

= K

by (iii). Hence S

⊆ K

. Moreover, by Lemma 2.2(iii) we have |K

| = |R

| = p

ϵκ

. Since

φ is a ring automorphism of R

, by Lemma 2.3(v) and κ

i,0

we have |S

| = |R|

i,0

= (p

ϵl

)

= |K

|. As stated above, we

conclude that K

= S

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