Let be a prime power and be the finite field with elements. Let be positive integers with . An linear code
is a subspace of the vector spacewith dimension . If this linear code is, in addition, closed under the cyclic shift, i.e., for any , then is called a cyclic code. Each vector is customarily identified with its polynomial representation , and a code is identified with the set of polynomial representations of its codewords. A linear code of length over is cyclic if and only if is an ideal of . It is well known that every ideal of is principal. Hence, there is a monic divisor of such that . The polynomial is called the generator polynomial of , and is called the parity-check polynomial of . If has irreducible factors over , we say such a cyclic code has nonzeros.
Suppose is a positive integer with . Let , i.e., the multiplicative order of modulo is , and be a primitive element in . Assume that and , then is a primitive -th root of unity. For each , let be the minimum polynomial of over . A cyclic code of length over is called a BCH code with designed distance if its generator polynomial is of the form
where lcm denotes the least common multiple of the polynomials, and . Denote such a BCH code with designed distance by . If it is called a narrow-sense BCH code and we denote it by . Clearly, . We denote by .
BCH codes were invented by Hocquenghem , and independently by Bose and Ray-Chaudhuri . One of the key features of BCH codes is a precise control over the number of symbol errors correctable by the code. Another advantage of BCH codes is that they have efficient encoding and decoding algorithms. Due to BCH codes have such good properties, they are widely used in DVDs, solid-state drives, compact disc players, disk drives, two-dimensional bar codes and satellite communications.
I-B Known Results
BCH codes have been extensively studied in the literature ([48, 14, 1, 2, 4, 5, 7, 6, 8, 9, 10, 11, 19, 18, 26, 23, 27, 25, 36, 28, 35, 40, 45, 46, 29, 39, 16, 20, 21, 22, 24, 33, 34, 37, 38, 42, 47]). Nonetheless, their parameters are known for only a few special classes. As pointed out by Charpin , the dimension and minimum distance of BCH codes are difficult to determine in general. The dimensions of the BCH codes were investigated in a lot of papers. We roughly list them in the Table I. Besides the results in Table I, for and , the dimension of was settled by Aly et al. . Recently, the dimensions of some BCH codes with were settled in [28, 29, 35].
The exact minimum distance of BCH codes has been studied in many literatures ( [7, 9, 10, 11, 23, 36, 14, 20, 34]). The reader is referred to  for a recent summary of various results on minimum distance of BCH codes. In general, the problem of determining the weight distribution of BCH codes is very difficult, and it is known for only a few special classes. Not much work has been done on determining the weight distribution of BCH codes. We list them in the following two cases.
Case 1: . For and , when , the weight distribution of was settled by Kasami ; when is a prime, the weight distribution of and was settled by Ding et al. . For and , the weight distribution of was determined by Yan . Recently, For , where , the weight distribution of was determined by Li .
Case 2: and . For , where , the weight distribution of and was settled by Li et al. .
I-C The contribution of the present paper
The objective of this paper is to study narrow-sense BCH codes over of length , where is a positive factor of . The main contributions are the following:
For and with , we give a trace representation for the codewords in and . By using exponential sums, the weight distribution of the BCH code and is settled. These results generalize those from .
For or for some integer , the first largest -cyclotomic coset leader modulo is determined, and then the weight distribution of a class of BCH codes of length is determined.
The paper is organized as follows. In Section II, we give some background and recall some basic results on character sums. By using cyclotomic cosets, the dimension of this class of narrow-sense BCH codes is determined in Section III. In Section IV, we find a trace representation for the codewords in and , where with . In addition, by using exponential sums and the theory of quadratic forms over finite fields, the weight distributions of and are determined. Moreover, the weight distribution of a class of BCH codes of length is also determined. Furthermore, a subclass of such BCH codes meeting the Griesmer bound is presented. Compared with the table of the best known linear codes maintained by Markus Grassl at http://www.codetables.de/, which is called the Database later in this paper, these two classes of BCH codes are sometimes among the best liner codes known. Finally, the conclusion of the paper is given in Section V.
Throughout this paper, let be a positive divisor of and , where is a positive integer. Clearly, and .
Let be a primitive element of and put , then is a primitive -th root of unity. For any , the -cyclotomic coset of modulo is defined as , where is the least positive integer such that and is the size of . Obviously, . The smallest element in is called the coset leader of . For every and , we define
Obviously, is the generator polynomial of . If , the dimension of is
Moreover, . The following is the well known BCH bound.
[40, Ch. 7, Th. 8] The minimum distance of is at least .
Let be the characteristic of , then is a power of . Let be the trace mapping from to and , where is a positive integer. For any given , the function is an additive character of . The character is called the canonical character of . Let be a fixed primitive element of . For each , the function with for defines a multiplicative character of , and every multiplicative character of can be defined in this way. The character is called the trivial multiplicative character of . When is odd, the character is called the quadratic character of , and is usually denoted by . Let be a multiplicative character and an additive character of . Then the Gaussian sum is defined by . From now on we shall denote the Gaussian sum over by . The explicit value of is known.
[32, Theorems 5.15, 5.33] Let , where is an odd prime and is a positive integer. Then
and for each ,
where is the quadratic character of .
We recall the following trace representation of cyclic codes, which is a direct consequence of Delsarte’s Theorem .
[36, Proposition 18] Let be a prime power and . Let be a primitive -th root of unity in and be a cyclic code of length over . Suppose has nonzeros and let be the roots of its parity-check polynomial which are not conjugate with each other. Denote the size of the -cyclotomic coset to be , . Then has the following trace representation
We give a brief introduction to the theory of quadratic forms over finite fields, which is used to calculate the weight distribution of BCH codes. Quadratic forms have been well studied ([15, 30, 31, 44, 49]). The form is called a quadratic form over if is a homogeneous polynomial of degree two in the form
If is odd, for a quadratic form in variables over , there exists a symmetric matrix of order over such that , where and denotes the transpose of . Let , then there exists such that is a diagonal matrix and , where . Let and assume that when . Let be the quadratic character of , then is an invariant of under the conjugate action of . We identify with the -dimensional -vector space. The following results are useful in the sequel.
Iii The dimension of BCH code of length
In this section, we will determine the dimension of the BCH codes for . Recall
Let . Then , since if there exists an integer with such that . Let denote the set of coset leaders in and the set of non coset leader in . Then . Note that if , there is an integer such that and is a coset leader of . That is, for every , there exists an integer such that . It follows that
Hence, to determine the dimension of the code , we need to find out the coset leader of and its cardinality for each .
The following result given in  will be useful for determining coset leaders when is small.
[3, Lemmas 8, 9] Let be an integer with , where . Then the -cyclotomic coset has cardinality for all in the range . Moreover, the following assert holds: every with in this range is a -cyclotomic coset leader modulo .
When is odd, by Lemma 5, we have the following conclusion.
Let be odd. For every integer with , has length , minimum distance and dimension .
Let . Let be an integer with and . Let , where are integers. Then is not a -cyclotomic coset leader modulo if and only if , where
We claim that an integer is not the coset leader in the -cyclotomic coset of modulo if and only if is not the coset leader in the -cyclotomic coset of modulo . In fact, is not a coset leader if and only if there exists an integer with such that , for some integer . Note that is equivalent to . Hence, the above assert holds.
We divide into two cases to prove our result.
If , an integer with and is not a coset leader if and only if , where , which has been proven in .
If , an integer with and is not the coset leader in the -cyclotomic coset of modulo if and only if is not the coset leader in the -cyclotomic coset of modulo . From Case (i), for integers .
If , then . Suppose and . Then, , where . That is, .
If , there exist integers such that , where and . Note that , thus, , where . Notice that . We claim . Otherwise, . We continue our discussions by distinguishing the following two subcases.
If , i.e., , then .
If , then . It gives . Hence, , where and . That is, .
The result follows. ∎
Define when is odd. Otherwise,
We have the following conclusion.
Let be defined as above. Let , and be an integer with . Then if and only if .
Clearly, divides . Notice that for each . Hence, if and only if
Let . Let be an integer with and . Then is not a -cyclotomic coset leader modulo if and only if , where , and are defined as Lemma 6. Moreover,
where is defined as above.
The following corollary can be deduced from Theorem 2.
Let , then
if is odd, the smallest with that is not a -cyclotomic coset leader modulo is ;
if , the smallest with that is not a -cyclotomic coset leader modulo is ;
if is even, the smallest with that is not a -cyclotomic coset leader modulo is .
Recall defined as Lemma 6. Let denote the smallest number in for . It is easy to check that has the following properties. If , then for all and . If , then for all . Hence, if , we have , and . This gives that the smallest with that is not a coset leader is if . Hence, the results of Cases (i) and (iii) are follow.
If and , then and . It follows that the smallest with that is not a coset leader is . If and , then . That is, every integer with is a coset leader. Notice that , we have that is not a coset leader. The proof is completed. ∎
For , the result that the smallest with that is not a a -cyclotomic coset leader modulo is was shown in . Moreover, for , the dimension of was determined in . For , if , the dimension of was determine in . Theorem 3 is a generalization of the results in . With the conclusions on cyclotomic cosets in Theorem 2, we determine the dimension of with as follows.
Let . For every integer with , let and , where . Then has length , minimum distance and dimension , where
if , define , then
if and , define , then
if and , then