BCH码的Bose距离与最小距离研究

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"一类BCH码的Bose和最小距离" 本文主要研究了一类特殊的循环码——Bose-Ray-Chaudhuri-Hocquenghem (BCH) 码的Bose距离和最小距离。BCH码是循环码的一个子类，因其高效的编码和解码算法在理论与实践中都具有重要意义，常被应用于通信系统、存储设备和消费电子产品中。尽管BCH码具有良好的错误校正能力，但其维度和最小距离通常未知。首先，定义一些基本概念。在数学领域，特别是编码理论中，一个长度为n的[k,d]码C是GF(q)（有限域）上的一个k维子空间，其中d表示码字之间的最小汉明距离。一个线性[n,k]码C称为循环码，如果它的任何码字(c0, c1, ..., cn-1)的循环移位形式(cn-1, c0, c1, ..., cn-2)仍属于C。 Bose距离，由Bose等人引入，是另一种衡量码字之间差异的距离度量。它与汉明距离类似，但考虑了码字的周期性质。在本文中，作者的目标是确定一类窄义原生BCH码的Bose距离和最小距离。窄义原生BCH码是一种特殊类型的BCH码，其生成多项式具有特定性质。 BCH码的构造通常基于生成多项式，该多项式定义了码字的循环性质。通过选取特定的生成多项式，可以确保码字能够纠正一定数量的突发错误，这在实际应用中非常关键。最小距离是码字间最小的汉明距离，决定了码的错误检测和纠正能力。对于BCH码，最小距离的计算涉及到伽罗华域的代数性质，包括伽罗华多项式的乘法和除法。在介绍完基本背景后，论文可能深入探讨了如何利用伽罗华域的理论来分析这类BCH码的生成多项式，并推导出其Bose距离的公式。此外，作者可能还提出了有效的算法或方法来计算这些码的实际最小距离，这对实际应用至关重要。通过这种方法，论文旨在填补理论上的空白，提供一种确定特定BCH码族的结构特性的方法。这样的研究对于优化编码方案，提升通信系统的可靠性和效率，以及设计更高效的数据存储系统具有重要意义。总结来说，这篇研究论文专注于一类BCH码的Bose距离和最小距离的计算，通过深入的代数分析和可能的算法开发，为理解和应用这类重要的线性循环码提供了新的理论基础。

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 5, MAY 2015 2351

The Bose and Minimum Distance of a

Class of BCH Codes

Cunsheng Ding, Senior Member, IEEE, Xiaoni Du, and Zhengchun Zhou

Abstract—Cyclic codes are an interesting class of linear

codes due to their efﬁcient encoding and decoding algorithms.

Bose-Ray-Chaudhuri-Hocquenghem (BCH) codes form a sub-

class of cyclic codes and are very important in both theory

and practice as they have good error-correcting capability and

are widely used in communication systems, storage devices, and

consumer electronics. However, the dimension and minimum

distance of BCH codes are not known in general. The objective

of this paper is to determine the Bose and minimum distances

of a class of narrow-sense primitive BCH codes.

Index Terms— BCH codes, cyclic codes, linear codes.

I. INTRODUCTION

HROUGHOUT this paper, let p be a prime and let

q be a power of p.An[n, k, d] code C over GF(q) is a

k-dimensional subspace of GF(q)

with minimum (Hamming)

distance d.

A linear [n, k] code C over GF(q) is called cyclic if

, c

, ···, c

n−1

) ∈ C implies (c

n−1

, c

, ···, c

n−2

) ∈ C.

By identifying any vector (c

, c

, ···, c

n−1

) ∈ GF(q)

with

+ c

x + c

+···+c

n−1

∈ GF(q)[x]/(x

− 1),

any code C of length n over GF(q) corresponds to a subset of

the quotient ring GF(q)[x]/(x

−1). A linear code C is cyclic

if and only if the corresponding subset in GF(q)[x]/(x

− 1)

is an ideal of the ring GF(q)[x]/(x

− 1).

Note that every ideal of GF(q)[x]/(x

− 1) is principal.

Let C =g(x) be a cyclic code, where g(x) is monic and

has the smallest degree among all the generators of C.Then

g(x) is unique and called the generator polynomial, and

h(x) = (x

− 1)/g(x) is referred to as the parity-check

polynomial of C.

Let m ≥ 1 be a positive integer and let n = q

− 1. Let

α be a generator of GF(q

)

∗

.Foranyi with 1 ≤ i ≤ q

−2,

Manuscript received May 9, 2014; revised October 19, 2014; accepted

February 6, 2015. Date of publication March 9, 2015; date of current version

April 17, 2015. C. Ding’s work was supported by the Research Grants

Council, University Grants Committee, Hong Kong, under Project 601013.

X. Du’s work was supported by the National Natural Science Foundation

of China under Project 61202395 and Project 61462077 and in part

by the Program for New Century Excellent Talents in University under

Grant NCET-12-0620. Z. Zhou’s work was supported in part by the Natural

Science Foundation of China under Project 61201243 and in part

by the Sichuan Provincial Youth Science and Technology Fund under

Grant 2015JQO004.

C. Ding is with the Department of Computer Science and Engineering,

The Hong Kong University of Science and Technology, Hong Kong (e-mail:

cding@ust.hk).

X. Du is with the College of Mathematics and Statistics, Northwest Normal

University, Lanzhou 730070, China (e-mail: ymldxn@126.com).

Z. Zhou is with the School of Mathematics, Southwest Jiaotong University,

Chengdu, 610031, China (e-mail: zzc@home.swjtu.edu.cn).

Communicated by J. Lahtonen, Associate Editor for Coding Theory.

Digital Object Identiﬁer 10.1109/TIT.2015.2409838

let m

(x) denote the minimal polynomial of α

over GF(q).

For any 2 ≤ δ<n = q

− 1, deﬁne

(q, m,δ)

(x) = lcm(m

(x), m

(x), ···, m

δ−1

(x)),

where lcm denotes the least common multiple of these

minimal polynomials. Let C

(q, m,δ)

denote the cyclic code

of length n with generator polynomial g

(q, m,δ)

(x).This

code C

(q, m,δ)

is called the narrow-sense primitive Bose-Ray-

Chaudhuri-Hocquenghem (BCH) code with design distance δ.

It is well known that the codes C

(q, m,δ)

and C

(q, m,δ



)

may be

equal for two different δ and δ



. The largest design distance

of a BCH code is called the Bose distance of the code and is

denoted by d

The codes C

(q, m,δ)

are treated in every book on coding

theory. However, the following questions about the codes

(q, m,δ)

are still open in general.

1) What is the dimension of C

(q, m,δ)

2) What is the Bose distance (i.e., the maximum design

distance) of C

(q, m,δ)

3) What is the minimum distance d (i.e., the minimum

weight) of C

(q, m,δ)

The dimension of C

(q, m,δ)

is known when δ is small, and

is open in general. There are lower bounds on the dimension

of C

(q, m,δ)

, which are very bad in many cases. The minimum

distance d of C

(q, m,δ)

is known only in a few cases. Only

when δ is very small or when C

(q, m,δ)

is the Reed-Solomon

code, both the dimension and minimum distance of C

(q, m,δ)

are known. Hence, we have very limited knowledge of the

narrow-sense primitive BCH codes, not to mention BCH codes

in general. Thus we conclude that BCH codes are neither

well-understood nor well-studied.

In the 1990’s, there were a few papers on narrow-sense

primitive BCH codes [1], [2], [5], [12]. However, in the last

eighteen years, little progress on the study of these codes has

been made. As pointed out by Charpin in [6], it is a well-

known hard problem to determine the minimum distance of

narrow-sense BCH codes.

The objective of this paper is to determine the Bose distance

and the minimum distance of the codes C

(q, m,δ)

with design

distance δ = q

+ h,where1≤ t ≤ m − 2and0≤ h ≤

(q

− 1)/q

m−t

+1.

II. K

NOWN RESULTS ABOUT THE CODES C

(q, m,δ)

To give a well-rounded treatment of the codes C

(q, m,δ)

we summarize known results on the dimension and minimum

distance of the codes in this section.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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