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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013
A Family of Five-Weight Cyclic Codes and Their
Weight Enumerators
Zhengchun Zhou, Cunsheng Ding, Senior Member, IEEE, Jinquan Luo, Member, IEEE, and Aixian Zhang
Abstract—Cyclic codes are a subclass of linear codes and
have applications in consumer electronics, data storage systems,
and communication systems as they have efficient encoding
and decoding algorithms. In this paper, a family of -ary cyclic
codes whose duals have three pairwise nonconjugate zeros is
proposed. The weight distribution of this family of cyclic codes is
determined. It turns out that the proposed cyclic codes have five
nonzero weights.
algorithms (see [12] for details). Thus, the study of the weight
distribution of a linear code is important in both theory and applications.
An
linear code
over
is called cyclic if
implies
.
By identifying any vector
with
Index Terms—Cyclic codes, exponential sum, quadratic form,
weight distribution, weight enumerator.
any code
I. INTRODUCTION
A
N
linear code over the finite field
is an -dimensional subspace of
with minimum (Hamming)
distance , where is a prime. Let
denote the number of
codewords with Hamming weight in a code of length .
The weight enumerator of is defined by
The sequence
is called the weight distribution
of the code. Clearly, the weight distribution gives the minimum
distance of the code and, thus, the error correcting capability.
In addition, the weight distribution of a code allows the computation of the error probability of error detection and correction with respect to some error detection and error correction
Manuscript received February 05, 2013; revised May 28, 2013; accepted
May 31, 2013. Date of publication June 11, 2013; date of current version
September 11, 2013. Z. Zhou was supported in part by the Natural Science
Foundation of China under Grant 61201243, in part by the Hong Kong Research
Grants Council under Grant 600812, in part by the application fundamental
research plan project of Sichuan Province, and in part by the Fundamental
Research Funds for the Central Universities under Grants SWJTU12CX053,
SWJTU12ZT15, and SWJTU12ZT14. C. Ding was supported by The Hong
Kong Research Grants Council, Project No. 600812. J. Luo was supported in
part by the Norwegian Research Council under Grant 191104/V30, in part by
the National Science Foundation (NSF) of China under Grant 60903036, in part
by the NSF of Jiangsu Province under Grant 2009182, and in part by the Open
Research Fund of the National Mobile Communications Research Laboratory,
Southeast University (No. 2010D12).
Z. Zhou is with the School of Mathematics, Southwest Jiaotong University,
Chengdu 610031, China, and also with the State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China (e-mail:
zzc@home.swjtu.edu.cn).
C. Ding is with the Department of Computer Science and Engineering, The
Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong (e-mail: cding@ust.hk).
J. Luo is with the Department of Informatics, University of Bergen, N-5020
Bergen, Norway (e-mail: jinquan.luo@ii.uib.no).
A. Zhang is with the Department of Mathematical, Xi’an University of Technology, Xi’an 710048, China (e-mail: zhangaixian1008@126.com).
Communicated by I. F. Blake, Associate Editor for Complexity and Cryptography.
Digital Object Identifier 10.1109/TIT.2013.2267722
of length over
corresponds to a subset of
. The linear code is cyclic if and only if the corresponding subset in
is an ideal. It is well known
that every ideal of
is principal. Let
,
where
is monic and has the least degree. Then,
is
called the generator polynomial and
is
referred to as the parity-check polynomial of . A cyclic code
is called irreducible if its parity-check polynomial is irreducible
over . Otherwise, it is called reducible. A cyclic code over
is said to have zeros if all the zeros of the generator polynomial of the code form conjugate classes, or equivalently, the
generator polynomial has irreducible factors over .
The weight distributions of both irreducible and reducible
cyclic codes have been interesting subjects of study for many
years. The determination of weight distributions is a hard
problem in general. For information on the weight distribution
of irreducible cyclic codes, the reader is referred to the recent
survey [4]. Information on the weight distribution of reducible
cyclic codes could be found in [5], [9], [15]–[17], [21], [23],
and [25].
For the duals of the known cyclic codes whose weight distributions were established, most of them have at most two zeros
(see [4], [5], [9], [10], [15]–[17], and [21]–[23]), only a few of
them have three or more zeros (see [9], [13], [15], and [25]).
The objective of this paper is to settle the weight distribution
of a family of five-weight cyclic codes whose duals have three
zeros.
This paper is organized as follows. Section II defines the
family of cyclic codes. Section III presents results on quadratic
forms which will be needed in subsequent sections. Section IV
solves the weight distribution problem for the family of cyclic
codes. Section V concludes this paper and makes some comments.
II. FAMILY OF CYCLIC CODES
In this section, we introduce the family of cyclic codes to be
studied in the sequel. Before doing this, we first give some notations which will be fixed throughout the paper unless otherwise
stated.
0018-9448 © 2013 IEEE
ZHOU et al.: FAMILY OF FIVE-WEIGHT CYCLIC CODES AND THEIR WEIGHT ENUMERATORS
Let
be an odd prime and
, where
is odd and
. Let
and
, where
is any positive integer with
. The conclusions in
the following lemma will be used in the sequel and their proofs
can be found in [18] and [20].
Lemma 2.1: With the notations above, we have
1)
;
2)
;
3)
; and
4)
for
.
Let be a generator of the finite field , and let
denote
the minimal polynomial of
over
for any integer . Note
that
by Lemma 2.1.
Thus,
, and
are irreducible polynomials
over
of degree . Since there does not exist any positive
integer such that
to
is called a quadratic form over
sented as
6675
if it can be repre-
where
. That is,
is a homogeneous polynomial
of degree 2 in the ring
.
The rank of the quadratic form
is defined as the codimension of the -vector space
That is,
, where is the rank of
.
In order to determine the weight distribution of the aforementioned code
, we need to deal with the exponential sum
of the following form:
(3)
for any
such that
, and there does not exist any positive integer
the elements
,
, and
are not the conjugate of each
other and have distinct minimal polynomials over
. So the
polynomials
, and
are distinct. Define
where is a complex primitive th root of unity, and
function from
to
satisfying
1)
for all
; and
2)
is a quadratic form over .
Note that any nonsquare in
is also a nonsquare in
is odd. It is easy to verify that
(1)
has degree
and is a factor of
.
Then,
Let
be the cyclic code with parity-check polynomial
. Then,
has length
and dimension
. Using
the well-known Delsarte’s Theorem [2], one can prove that
(2)
where the codeword
and
denotes the absolute trace from
to .
Let
and
be the cyclic code
. Then,
is a subwith parity-check polynomial
code of
with dimension
. Trachtenberg [20] proved
that
has three nonzero weights and determined its
weight distribution. The objectives of this paper are to show
that
has five nonzero weights and settle the weight
distribution of this class of cyclic codes.
III. MATHEMATICAL FOUNDATIONS
In this section, we give a brief introduction to quadratic forms
over finite fields which will be useful in the sequel. Quadratic
forms have been well studied (see the monograph [14] and the
references therein) and have applications in sequence design
(see [11], [19], and [20]), and coding theory (see [9], [15], [16],
and [25]).
Definition 3.1: Let
, where
and
is a basis for
over . A function
from
is a
since
(4)
where is a fixed nonsquare in . The following result can be
traced back to Trachtenberg [20] whose proof is based on (4)
and the classification of quadratic forms over finite fields in odd
characteristic. For more details, we refer the reader to [20, pp.
30–36] and [19, Lemma 4].
Lemma 3.2: Let
be defined by (3) and be the rank of the
quadratic form
. Then,
if is odd, and
otherwise.
IV. WEIGHT DISTRIBUTION OF THE FAMILY OF CYCLIC CODES
In this section, we shall establish the weight distribution of
the code
of (2) defined in Section II. To this end, we
need a series of lemmas. Before introducing them, for any
, we define
(5)
and
(6)
be defined by (5). Then,
Lemma 4.1: Let
for any
. And for any
, the quadratic
form
has rank
for some
.
Proof: Recall that
,
and
for
. Thus,
and
for any
. This together with the linear properties of the
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trace function means that
Note that
where
,
over . Thus,
calculate the rank of
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013
for any
.
and
is a basis of
is a quadratic form over . We now
. Note that
TABLE I
VALUE DISTRIBUTION OF
Case B, when
: In this case, for each
equation system (9) has the same number of solutions
as
, the
where
which has the same number of solutions
as
We need to calculate the number of roots of the linearized polynomial
. Let
. Then
(10)
Squaring both sides of each equation in (10), we have
(7)
.
Clearly,
has the same number of roots in
as
Fix an algebraic closure
of ; then, all roots of
form a vector space over
of dimension at most 4 since its
degree is at most
for any
.
Note that
, it is straightforward (see [20, Lemma
4]) to verify that elements in
that are linearly independent
over
are also linearly independent over
. Therefore, the
roots of
in
form a vector space over
of dimension at most 4. Thus, the rank of
is at least
for any
. This completes the proof.
Lemma 4.2: Let
denote the number of solutions
of the following system of equations:
which implies that
since
. Conversely, for any
, it is clear that is a solution to
(10) since
and
for each
.
Thus, (10) has exactly solutions.
Summarizing the results of the two cases above, we have that
. This completes the proof.
The following lemma is the key to establishing the weight
distribution of the proposed code
. Its proof is lengthy
and is presented in the Appendix.
Lemma 4.4: Let
denote the number of solutions
of the following system of equations:
(8)
(11)
Then,
Proof: The conclusion follows directly from the observation that
is a solution of (8) if and only if
.
Lemma 4.3: Let
denote the number of solutions
of the following system of equations:
(9)
Then,
Proof: We distinguish between the following two cases to
calculate the number of solutions
of (9).
Case A, when
: In this case, by Lemma 4.2, the number
of solutions of (9) is equal to .
Then,
Proof: See the Appendix.
Theorem 4.5: Let
be defined by (6). Then, as
runs
through , the value distribution of
is given by Table I.
Proof: It is clear that
if
. Otherwise, by Lemmas 4.1 and 3.2, we have
To determine the distribution of these values, we define
ZHOU et al.: FAMILY OF FIVE-WEIGHT CYCLIC CODES AND THEIR WEIGHT ENUMERATORS
where
. Then, the value distribution of
is as follows
TABLE II
WEIGHT DISTRIBUTION OF THE CODE
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IN
THEOREM 4.6
(12)
where
immediately have
and
. By (12), we
Combining (13) and (14) gives
(13)
On the other hand, we have
The value distribution of
depicted in Table I then follows
from the values of
, and
, and the analysis
above.
The following is the main result of this paper.
Theorem 4.6: Let
be the code in (2). Then,
is a cyclic code over
with parameters
and
where
and
is the number of solutions of (8). Similarly, we have
Furthermore, the weight distribution of
is given by
Table II.
Proof: The length and dimension of the code follow directly from the definition of
. We only need to determine
its minimal weight and weight distribution. In terms of exponential sums, the weight of the codeword
in
is given by
and
where
and
are the number of solutions of (9) and (11),
respectively. Applying Lemmas 4.2, 4.3, and 4.4, we have
(14)
(15)
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013
TABLE III
WEIGHT DISTRIBUTION OF THE CODE
IN
THEOREM 5.1
where
is given by (6), and in the fifth identity, we used the
fact that
for any
and
. The minimal
weight and weight distribution of
then follow from (15)
and the value distribution of the exponential sum
depicted
in Table I.
Example 4.7: Let
,
, and
. Then, the
code
is a
code over
with the weight
enumerator
which agrees with the weight distribution in Table II.
Example 4.8: Let
,
and
. Then, the
code
is a
code over
with the weight
enumerator
Finally, we would mention some applications of the codes
with five weights considered in this paper. Cyclic codes with
a few weights are of special interest in authentication codes as
certain parameters of the authentication codes constructed from
these cyclic codes could be easily computed [7], and in secret
sharing schemes as the access structures of the secret sharing
schemes derived from such cyclic code can be easily determined
[1], [6], [24]. Cyclic codes with a few weights are also of special
interest in designing frequency hopping sequences [3], [8].
APPENDIX A
PROOF OF LEMMA 4.4
(16)
It is then obvious that
For any
lutions
which agrees with the weight distribution in Table II.
Example 4.9: Let
,
, and
. Then, the
code
is a
code over
with the weight
enumerator
, let
denote the number of soof the following system of equations:
For any
lutions
, let
denote the number of soof the following system of equations:
(17)
Since
and
are odd,
, we have
(18)
which agrees with the weight distribution in Table II.
V. SUMMARY AND CONCLUDING REMARKS
In this paper, we studied a family of five-weight cyclic codes.
The duals of the cyclic codes have three zeros. The weight distribution of this family of cyclic codes is completely determined.
We mention that the weight distribution of
can also
be settled in a more general case where
is odd. In
what follows we only report the conclusion. The proof is similar
to that of Theorem 4.6.
Theorem 5.1: Let
,
be odd, and
.
Let
be the code in (2). Then,
is a cyclic code
over
with parameters
Furthermore, the weight distribution of
Table III.
is given by
We distinguish among the following three cases to calculate
.
Case A,
: In this case,
since
is a solution of (17) if and only if
. Thus,
.
Case B,
, and (
or
): In this case, without
loss of generality, we suppose that
. By Lemma 2.1,
, and thus,
is odd . It then follows
from the second equation in (17) that
. This leads to
which means that the first equation in (17) cannot
hold since
. Therefore,
in this case.
Case C,
,
and
. In this case, for any given
, equation system (17) has the same number of solutions
as
(19)
ZHOU et al.: FAMILY OF FIVE-WEIGHT CYCLIC CODES AND THEIR WEIGHT ENUMERATORS
where
as
and
. Clearly,
does. By Lemma 5.2, we have
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Thus
is uniquely determined by . Substituting (22) into
the second equation of (21), we obtain
runs over
(23)
Let
. Then, (23) is equivalent to
(24)
is not a square in , then
If (24) has no solution, i.e.,
. Otherwise, suppose that
and
are two
solutions of (24). We then have
Summarizing all the cases above, we have
This completes the proof.
Lemma 5.2: Let
denote the number of solutions
of (19), where
. Then, we have
the following conclusions.
B1
.
B2 When
runs over
,
for
cases
for
cases
in the remaining cases.
The proof of Lemma 5.2 is lengthy and technical. We first
prove some auxiliary results.
APPENDIX B
AUXILIARY RESULTS FOR PROVING LEMMA 5.2
(25)
or
(26)
Clearly, (25) and (26) have the same number of solutions
. Note that
. Thus, both (25) and
(26) have no solution or exactly
solutions. If
,
then
and
. However
is not a square, thus,
and
. In this case, (25) and (26) become the same
equation and have
solutions. If
, then (25) and
(26) have distinct solutions.
Based on the above analysis, we conclude
Define
.
We prove Lemma 5.2 only for the case that
The proof for the case
is similar and omitted.
Hence, we assume that
from now on.
A) Case 1: In (19), we substitute
with
and
obtain the following system of equations:
Note that the first equation in (21) has
solutions in
thanks to [14, Lemma 6.24]. When
runs through all these
solutions, the second equation in (21) will give a
-to-1
correspondence
(20)
. Our task is to compute the number
of
where
solutions
of (20). To this end, we first compute the
number
of solutions
of the following system
of equations:
if
. Therefore
which leads to
(21)
where
.
Lemma 5.3: Let symbols and notations be the same as before. As for (21), we have
if
for
elements
for the remaining
Proof: Let
(21). It is clear that
that
and
This completes the proof.
Lemma 5.4: Let symbols and notations be the same as before. As for equation system (20), we have
if
for
pairs
for the remaining
.
be a solution of the first equation in
. Let
. It then follows
(22)
.
be any solution of (20). Let
Proof: Let
. It then follows from the first equation in (21) that
(27)
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013
It then follows from Lemma 5.4 that
Using the second and third equations in (20), we obtain
and
Let
and
for
pairs
for the remaining
(28)
Then,
.
The proof is then completed.
B) Case 2:
Lemma 5.6: Let
denote the number of solutions
of (19) such that is a square and is a nonsquare or
. Then
and
It follows from the first equation in (28) that
if
for
pairs
for the remaining
(29)
Combining the first equation in (28) and (29), we obtain
.
This case is symmetric to Case 1. Hence, the proof of this
lemma is similar to that of Lemma 5.5 and is omitted.
with
and
C) Case 3: In (19), we substitute
obtain the following system of equations:
Hence,
(30)
(31)
Note that
. Our task is to compute the number
of
where
solutions
of (31). To this end, we first compute the
number
of solutions
of the following system
of equations:
if and only if
(32)
By (30), is uniquely determined by . Therefore, is uniquely
determined by .
In addition, it is easily seen that
if and only if
.
Hence the number of solutions of (20) is the same as that of
(21). The desired conclusions then follow from Lemma 5.3.
Lemma 5.5: Let
denote the number of solutions
of (19) such that is a square and is a nonquare or
. Then
if
for
pairs
for the remaining
.
where
Lemma 5.7: Let symbols and notations be the same as before. As for (32), we have
if
for
elements
for the remaining
Proof: Choose
such that
.
. From
(33)
we can assume
(34)
.
is a
Proof: Consider now the solutions of (20). If
solution of (20), so are
,
and
.
If
, there are indeed four different solutions of (20), but
they give only one solution of (19).
Since
is a quadratic nonresidue in ,
. However,
it is possible that
. If
, then
. In this
case, we have two special solutions
of (20). They give
only one solution of (19).
with
. It is easy to see that all the solutions
of (33) can be expressed as in (34) with a unique
Substituting (34) into
.
(35)
we obtain
(36)
ZHOU et al.: FAMILY OF FIVE-WEIGHT CYCLIC CODES AND THEIR WEIGHT ENUMERATORS
Denote by
if
. Then, (36) is equivalent to
6681
. Therefore
(37)
Let
and
have
.
From (34) and
be the two solutions of (37). Then, we
, we have
The proof is now finished.
Lemma 5.8: Let symbols and notations be the same as before. As for equation system (31), we have
which implies
• If
, then
. Since is odd,
. It follows that
. This implies that
. For a fixed , recall that
and
are the
two solutions of (37). Then, we have
(38)
or
(39)
If
which implies
are the two solutions of (38), then
which is equivalent
to
, since
due
to Lemma 2.1. As a consequence, if (38) has solutions,
then it has exactly
solutions.
If
, then (39) and (38) have the same solutions.
In this case
and
. But
can be
excluded since, otherwise,
, then
which contradicts to
. The remaining case is
which corresponds to
. In this case, we have
solutions of which gives exactly the same number
of solutions of (32).
If
, then (39) has the same number of solutions
as (38) and moreover, their solutions are distinct. Therefore, (38) and (39) both have
solutions or no solutions
in
.
• If
and
, then
. Note that
,
if
for
pairs
for the remaining pairs
.
Proof: The proof of this lemma is similar to that of Lemma
5.4 and is derived from Lemma 5.7. The details of the proof are
omitted here.
Lemma 5.9: Let
denote the number of solutions
of (19) such that both and are squares. Then
if
for
pairs
for the remaining
.
and
is not in
except for
the exception case will not occur since
Summarizing up, we conclude
. But
.
Define
Note that (33) has
solutions in
thanks to in [14, Lemma
6.24]. When
runs through all these solutions, (35) will
give a
-to-1 correspondence
Proof: Consider now the solutions of (31). If
is a
solution of (31), so are
,
, and
.
If
, there are indeed four different solutions of (31),
but they give only one solution of (19).
However, it is possible that
. If
, (31)
has four special solutions
and
. They give only
two solutions of (19). It then follows from Lemma 5.8 that
If
, then the four distinct solutions
give only one solution of (19). In this case, it then follows from
Lemma 5.8 that
for
pairs
for the remaining
.
The proof is then completed.
D) Case 4:
Lemma 5.10: Let
denote the number of solutions
of (19) such that both and are either nonsquares or
zero. Then
if
for
pairs
for the remaining
.
Proof: The proof of this lemma is similar to that of Lemma
5.9 and is omitted here.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013
APPENDIX C
PROOF OF LEMMA 5.2
and
of (19) are counted
Note that the solutions
more than once in Cases 1–4. By analyzing the proofs of
Lemmas 5.5, 5.6, 5.9, and 5.10, we have
When
,
is the sum of the solutions given
in Lemmas 5.5, 5.6, 5.9, and 5.10. This completes the proof.
ACKNOWLEDGMENT
The authors are very grateful to the reviewers and the Associate Editor, Prof. Ian F. Blake, for their comments and suggestions that improved the presentation and quality of this paper. Z.
Zhou would like to thank Prof. X. Tang for his encouragement.
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Zhengchun Zhou received the B.S. and M.S. degrees in mathematics and
the Ph.D. degree in information security from Southwest Jiaotong University,
Chengdu, China, in 2001, 2004, and 2010, respectively. From 2012 to 2013,
he was a postdoctoral member in the Department of Computer Science and
Engineering, the Hong Kong University of Science and Technology. He is
currently an associate professor with the School of Mathematics, Southwest
Jiaotong University. His research interests include sequence design and coding
theory.
Cunsheng Ding (M’98–SM’05) was born in 1962 in Shaanxi, China. He received the M.Sc. degree in 1988 from the Northwestern Telecommunications
Engineering Institute, Xian, China; and the Ph.D. in 1997 from the University
of Turku, Turku, Finland.
From 1988 to 1992 he was a Lecturer of Mathematics at Xidian University,
China. Before joining the Hong Kong University of Science and Technology in
2000, where he is currently Professor of Computer Science and Engineering,
he was Assistant Professor of Computer Science at the National University of
Singapore.
His research fields are cryptography and coding theory. He has coauthored
four research monographs, and served as a guest editor or editor for ten journals.
Dr. Ding co-received the State Natural Science Award of China in 1989.
Jinquan Luo (M’09) was born in February 1980, Anhui, China. He received the
B.S. degree from Zhejiang University, Hangzhou, China, in July 2001 and the
Ph.D. degree from Tsinghua University, Beijing, China, in January 2007, both in
mathematics. He joined Yangzhou University, China, in 2007 and later become
a Research Fellow at Nanyang Technological University, Singapore from 2009.
Currently, he servers as postdoctor at the Department of Informatics, University
of Bergen, Norway. His major research interests are coding theory, cryptology,
and number theory.
Aixian Zhang received the M.S degree and the Ph.D degree in mathematics
from the Capital Normal University, Beijing, China, in 2010 and 2013, respectively. She is currently a Lecturer of Xi’an University of Technology. Her current research interests are coding theory and algebraic number theory.