Narrow-Sense BCH Codes over GF(q) with Length n= qm 1

Transcription

1 Narrow-Sense BCH Codes over GFq) with Length n= qm 1 Shuxing Li, Cunsheng Ding, Maosheng Xiong, and Gennian Ge 1 Abstract arxiv: v [cs.it] 1 Nov 016 Cyclic codes are widely employed in communication systems, storage devices and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over GFq) with length n=q m 1)/). Little is known about this class of BCH codes when q>. The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length n= m 1)/ are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated. Index Terms BCH codes, Bose distance, cyclic codes, minimum distance, quadratic forms, secret sharing, weight distribution. I. INTRODUCTION Throughout this paper, let q be a power of a prime p. An [n,k,d] linear code C over GFq) is a k-dimensional subspace of GFq) n with minimum Hamming distance d. If in addition of being a linear code, it satisfies the condition that any c 0,c 1,,c n 1 ) C implies c n 1,c 0,c 1,,c n ) C, then C is called a cyclic code. By identifying each vector c 0,c 1,,c n 1 ) GFq) n with a polynomial c 0 + c 1 x+c x + +c n 1 x n 1 R := GFq)[x]/x n 1), a linear code C of length n over GFq) corresponds to a GFq)-submodule of the ring R. Moreover, C is cyclic if and only if the corresponding submodule is an ideal of R. Note that every ideal of R is principal. Let C R be a cyclic code of length n over GFq), then C = gx) where gx) GFq)[x] may be chosen to be monic and to have the smallest degree among all the generators of C. This gx) is unique and satisfies gx) x n 1) and is called the generator polynomial, and hx) := x n 1)/gx) is called the parity-check polynomial of C. If the generator polynomial gx) resp. the parity-check polynomial hx)) can be factored into a product of s irreducible polynomials over GFq), then C is called a cyclic code with s zeroes resp. s nonzeroes). In this paper, we consider only cyclic codes of length n over GFq) with gcdn,q)=1, which implies that the generator polynomial of the code does not have repeated roots. The research of C. Ding was supported by the Hong Kong Research Grants Council, under Grant No The research of M. Xiong was supported by RGC grant number from Hong Kong. The research of G. Ge was supported by the National Natural Science Foundation of China under Grant Nos and S. Li is with the School of Mathematical Sciences, Zhejiang University, Hangzhou 1007, Zhejiang, China sxli@zju.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, China cding@ust.hk). M. Xiong is with the Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong mamsxiong@ust.hk). G. Ge is with the School of Mathematical Sciences, Capital Normal University, Beijing 10004, China gnge@zju.edu.cn). He is also with Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 10004, China.

2 Let n be a positive integer. Let m=ord n q), that is, m is the smallest positive integer such that n q m 1. Let α be a generator of GFq m ) := GFq m )\{0}, and put β=α qm 1)/n. Then β is a primitive n-th root of unity in GFq m ). For each i where 0 i n 1, let m i x) denote the minimal polynomial of β i over GFq). For each δ n, define g n,q,m,δ) x)=lcmm 1 x),m x),,m δ 1 x)), where lcm denotes the least common multiple of the polynomials. We also define g n,q,m,δ) x)=x 1)g n,q,m,δ) x). Let C n,q,m,δ) and C n,q,m,δ) denote the cyclic codes of length n with generator polynomials g n,q,m,δ) x) and g n,q,m,δ) x), respectively. Then C n,q,m,δ) is called a narrow-sense BCH code with designed distance δ, and C n,q,m,δ) is the even-like subcode of C n,q,m,δ). Clearly, we have dim C n,q,m,δ) )=dimc n,q,m,δ) ) 1. Since g n,q,m,δ) x) has δ 1 consecutive roots β i for all 1 i δ 1, and g n,q,m,δ) x) has δ consecutive roots β i for all 0 i δ 1, it follows from the BCH bound that the minimum distances of C n,q,m,δ) and C n,q,m,δ) are at least δ and δ+1, respectively. Due to this fact, δ is called the designed distance of the code C n,q,m,δ). It is well known that for two different design distances δ and δ, the codes C n,q,m,δ) and C n,q,m,δ ) may be the same. The largest designed distance of C n,q,m,δ) is called the Bose distance [5, p. 05] and is denoted by d B. Since the Bose distance of C n,q,m,δ) serves as the largest lower bound on the minimum distance of the code among all designed distances, it is useful to determine the Bose distance of the BCH code, if the minimum distance cannot be obtained. The cyclic codes C n,q,m,δ) are treated in almost every book on coding theory. When n=q m 1, the codes C n,q,m,δ) are called narrow-sense primitive BCH codes and have been extensively studied in the literature [], [], [4], [5], [], [9], [10], [1], [17], [19], [0], [1], [6], [7], [9], [1], []. The reader is referred to [17] for a recent summary of various results on narrow-sense primitive BCH codes. When n= qm 1, the codes C n,q,m,δ) and C n,q,m,δ) are also interesting, however, very little is known about them when q>. The objective of this paper is to study some special classes of the narrow-sense BCH codes of length q m 1 for q>. We employ various tools, including cyclotomic cosets, locator polynomials, nondecreasing sequence decompositions, exponential sums and the theory of quadratic forms over finite fields to study important parameters such as dimensions, Bose distances, minimum distances and weight distributions of these codes. A class of ternary BCH codes meeting the Griesmer bound is presented. Moreover, an application of some of the BCH codes presented in this paper is also investigated. As will be shown, some narrow-sense BCH codes of length qm 1 have optimal parameters. To investigate the optimality of some of the codes studied in this paper, we compare them with the tables of the best known linear codes maintained by Markus Grassl at which are called the Database later in this paper. In some cases, we also use the tables of the best cyclic codes given in the monograph [1] as benchmarks. II. PRELIMINARIES In this section, we present some necessary background concerning cyclotomic cosets, coset leaders, nondecreasing sequence decompositions, quadratic forms over finite fields and exponential sums. Some known results about BCH codes are also recalled.

3 A. Cyclotomic cosets Let n be a positive integer such that gcdn,q)=1. Let Z n denote the ring of integers modulo n. Let s be an integer with 0 s<n. The q-cyclotomic coset of s modulo n is defined by C s ={s,sq,sq,,sq l s 1 } mod n Z n, where l s is the smallest positive integer such that q l s s s mod n). l s is also the size of the q-cyclotomic coset C s. The smallest nonnegative integer in C s is called the coset leader of C s. Let Γ n,q) be the set of all the coset leaders. Then we have C s C t = /0 for any distinct elements s and t in Γ n,q), and s Γ n,q) C s =Z n. 1) Namely, the q-cyclotomic cosets modulo n form a partition of Z n. Suppose m=ord n q) and β is a primitive n-th root of unity in GFq m ). Then, the minimal polynomial m s x) of β s over GFq) is the monic polynomial of the smallest degree over GFq) with β s as a zero. By Galois theory, this polynomial is irreducible over GFq) and is given by It follows from 1) that m s x)= i C s x β i ) GFq)[x]. x n 1= s Γ n,q) m s x), which is the factorization of x n 1 into irreducible factors over GFq). Thus, for a cyclic code C = gx) GFq)[x]/x n 1), the generator polynomial gx) is a product of some m s x) s. Hence, the degree of gx) and the dimension of C can be determined by the size of cyclotomic cosets associated with gx). Regarding the size of q-cyclotomic cosets modulo n, we have the following lemma. Lemma 1. [1, Theorem 4.1.4] The size l s of each q-cyclotomic coset C s is a divisor of ord n q), which is the size l 1 of C 1. The following lemma says when s is small, the size of the q-cyclotomic coset C s Z n is always equal to ord n q). Lemma. [, Lemma ] Suppose q m/ < n q m 1, where m=ord n q). Then the q-cyclotomic coset C s has cardinality m for all s in the range 1 s nq m/ /q m 1). As a direct consequence, the dimension of some BCH codes can be easily obtained. Theorem. [, Theorem 10] Suppose q m/ < n q m 1, where m=ord n q). Then the narrow-sense BCH code C n,q,m,δ) with δ in the range δ min{ nq m/ /q m 1),n} has dimension k=n m δ 1)1 1/q). The following proposition illustrates the close relation between coset leaders and the Bose distance of the narrow-sense BCH code C n,q,m,δ). Proposition 4. The Bose distance d B of the code C n,q,m,δ) is a coset leader of a q-cyclotomic coset modulo n. Moreover, if δ is a coset leader, then d B = δ. Proof: Suppose d B is the Bose distance of C n,q,m,δ), then by definition, β i is a root of g n,q,m,δ) x) for each 1 i d B 1. Assume that d B is not a coset leader. Then there exists some 1 j d B 1,

4 4 such that d B C j. Therefore, β d B is also a root of g n,q,m,δ) x). This implies C n,q,m,δ) = C n,q,m,db +1) and C n,q,m,δ) has a designed distance d B + 1>d B. This contradicts to the definition of the Bose distance. Moreover, if δ is a coset leader, then, δ C j. 1 j δ 1 By the definition of g n,q,m,δ) x), β δ is not a root of g n,q,m,δ) x). Hence, δ is the largest designed distance of C n,q,m,δ) and we have d B = δ. B. Nondecreasing sequence decompositions and coset leaders There have been some interesting results on coset leaders of q-cyclotomic cosets when n=q m 1 [17], [6], [1]). Particularly, in [1], the concept of nondecreasing sequence decompositions was proposed and its close relation with coset leaders modulo q m 1 was discussed. In this subsection, we show that this concept is also helpful in determining some special q-cyclotomic coset leaders modulo qm 1. Here we always assume that q> is a prime power. Suppose v=v n 1,v n,...,v 0 ) is a sequence of length n with each component v i satisfying 0 v i. The sequence v is called a nondecreasing sequence NDS) if v i+1 v i for 0 i n. Every sequence has a unique nondecreasing sequence decomposition as a concatenation V 1 V...V r where the V i s are NDSs and r is minimal. Let V =v l,...,v 0 ) and W =u k,...,u 0 ) be two NDSs. We say V = W if l = k and v l i = w l i for 0 i l. We say V > W, if either l > k and v l i = w k i for 0 i k or there is an integer 0 j min{l,k} such that v l j > w k j and v l i = w k i for 0 i j 1. For two sequences of the same length with the NDS decomposition given by v= V 1 V...V r and w= W 1 W...W s, we say v>w if there is an integer 1 j min{r,s} such that V j > W j and V i = W i for 1 i j 1. We remark that the notion v>w described above is consistent with the natural inequality i v i q i > i w i q i of the corresponding integers. Given a positive integer s, we may assume that 0<s<n. Suppose the unique q-adic expansion of s is i=0 s iq i, where 0 s i. This defines the sequence s=s,s m,...,s 0 ). Denote by Es) the NDS decomposition of s. Conversely, let V 1 V...V r be the NDS decomposition of s. Then define E 1 V 1 V...V r )=s. The coset leader of C s modulo n is denoted by s. When n=q m 1 and s=s,s m,...,s 0 ), noting that n=,,,), it is clear that the sequence q i s corresponding to q i s mod n is q i s=s i,...,s 0,s,...,s m i ), 1 i. ) That is, when n=q m 1, multiplying a power of q simply corresponds to a cyclic shift of the original sequence s. This key fact is fundamental for the important results presented in [1]. When n= qm 1, the situation is more complicated: while ) is still true, however, when q i s>n, noting that n=1,1,...,1), we need to subtract q i s by some multiple of 1,1,...,1) so that the resulting sequence lies in between 0 and n. With this observation, we can translate easily some results of [1] into the new context, where the modulus is n= qm 1. From now on, we always consider q-cyclotomic cosets modulo n= qm 1. Lemma 5. Let 0 s qm 1 1 be an integer such that the components of s are either 0 or 1. Suppose Es)= V 1 V...V r. Then we have the following. i) Es )= V j V j+1...v r V 1...V j 1 for some j where V j V i for each 1 i r. ii) If V 1 = V = = V r or V 1 = V = = V j < V k for some j and for all k> j, then s=s. iii) If r = 1, then s=s.

5 5 Proof: Since n=1,1,,1) and the components of s are either 0 or 1, the components of q i s are also either 0 or 1. Thus 0<q i s<n for all i. Therefore the process of subtracting q i s by multiples of n is not involved. The proof is exactly the same as that of [1, Theorem.] for n=q m 1. Let v=v l 1,v l,...,v 0 ) be an NDS. Define the truncating operator T k as T k v)=v l 1,v l,...,v l k ), 1 k l. We now define the successor operator S in the way that Sv) is the smallest NDS that is larger than v. In particular, if v 0 <, we have Sv)=v l 1,v l,...,v 0 + 1), which corresponds to the successor of the integer l 1 i=0 v iq i. The following Lemma provides information about the coset leaders modulo qm 1 in a special case. Lemma 6. Let 0 s qm 1 1 be an integer. Suppose Es)= V 1V...V r, where V 1 > V. Suppose V 1 has length l and has components either 0 or 1. Let Ms) be the smallest coset leader greater than or equal to s. Write m=al+ b, where 0 b l 1. If b=0, we have If 1 b l 1, we have Ms)=E 1 V 1 V 1...V 1 ). }{{} a Ms) E 1 V 1 V 1...V 1 ST b V 1 ))). }{{} a In particular, if the last component of ST b V 1 )) is 1, then the equality holds. Proof: The proof is the same as that of [1, Theorem.5] for n=q m 1, because in ) no subtracting of 1,1,...,1) from q i s is involved. C. Gauss and exponential sums related to quadratic forms In this subsection, we list two results on Gauss sums and exponential sums related to quadratic forms over finite fields. Interested readers may refer to [, Chapter 5] for other properties of Gauss sums and to [, Chapter 6] and [] for the general theory of quadratic forms over finite fields. From what follows, we assume that p is an odd prime and q = p s. Denote by Tr the standard trace map from GFq) to GFp). Definition 1. Let χ be a multiplicative character over GFq). The Gauss sum Gχ) is defined to be Gχ) = χx)ζ Trx) p, x GFq) where ζ p := expπ 1/p) is a p-th complex root of unity. Let η be the quadratic character of GFq). Lemma 7. [, Theorem 5.15] The quadratic Gauss sum Gη) satisfies { 1) s 1 q if p 1 mod 4, Gη)= 1) s 1 1) s q if p mod 4. The following identity also holds see [, Theorem 5.]): ζ Trax) p = ηa)gη) 1, a GFq). ) x GFq)

6 6 Lemma. [, Lemma 1] Let Qx) be a quadratic form from GFq m ) to GFq) with rank r. Then { ζ TrQx)) ±q m r/ if q 1 mod 4, p = x GFq m ) ± 1) r q m r/ if q mod 4. Moreover, for a GFq), we have D. Some known results concerning BCH codes ζ TraQx)) p = ηa r ) ζ TrQx)) p. x GFq m ) x GFq m ) Locator polynomials are very useful in the study of BCH codes [], [4]. We first review the definition. Definition. Let c=c 0,c 1,...,c n 1 ) GFq) n be a vector with nonzero components c i1,c i,...,c iw. Then X 1 = β i 1,...,X w = β i w, are called the locators of c, where β is a primitive n-th root of unity in GFq m ). The locator polynomial of c is σz)= w i=1 1 X i z)= w i=0 σ i z i, where σ 0 = 1. The coefficients σ i are the elementary symmetric functions of X i : σ 1 = X 1 + +X w ), σ = X 1 X + X 1 X + +X w 1 X w,. σ w = 1) w X 1 X w. The following lemma suggests a method to find a codeword of prescribed weight in the BCH code C n,q,m,δ) by using locator polynomials. Lemma 9. [5, Ch. 9, Lemma 4] Let σz)= w i=0 be a polynomial over GFq m ). Then σz) is the locator polynomial of a codeword c C n,q,m,δ) with only 0 and 1 components if and only if the following two conditions hold. i) The zeroes of σz) are distinct n-th roots of unity. ii) σ i = 0 for all 1 i δ 1 with p i, where p is the characteristic of GFq). The following lemma says in some cases, the minimum distance equals the designed distance. Lemma 10. [6, Theorem 4..1] For the BCH code C n,q,m,δ), if δ n, then the minimum distance d = δ. III. THE NARROW-SENSE BCH CODES WITH LARGE DIMENSIONS For the rest of this paper, unless otherwise stated, we will always assume that n = qm 1. Therefore ord n q)=m. We use α to denote a primitive element of GFq m ) and β=α. In this section, we consider the narrow-sense BCH codes C n,q,m,δ) and C n,q,m,δ) with few zeroes, and thus they have large dimensions. The narrow-sense BCH code with designed distance has only one zero. The parameters of these codes are known. σ i z i

7 7 Theorem 11. The code C n,q,m,) has parameters [q m 1)/),q m 1)/) m,d], where { if gcdm,)= 1, d = if gcdm,)> 1. Proof: The dimension follows from the fact that C 1 =m. Since C n,q,m,) has only one zero β, its parity-check matrix is H =1,β,...,β n 1 ), where the i-th column is a vector in GFq) m corresponding to β i 1. Notice that gcdn,)=gcdq + q m + +q+1,)=gcdm,). If gcdm, q 1) = 1, then gcdn, q 1) = 1. Suppose H contains two columns which are linearly dependent over GFq). Let these two columns be the i-th column and the j-th column where 1 i< j n. Then we have β j 1 = θβ i 1 for some θ GFq). Thus β j i = θ GFq), which implies that β j i)) = 1. Since β is a primitive n-th root of unity and gcdn,) = 1, we have n j i). This is impossible since 1 j i n 1. Therefore every two distinct columns of H are linearly independent over GFq) if gcdm,)=1. In this case the code C n,q,m,) is the Hamming code and we have d = [1, p. 9-0]. If gcdm,)=l > 1, from the above argument it is easy to see that the first column and the n l +1)-th column of H are linearly dependent over GFq). That is, the code C n,q,m,) has minimum distance d =. Theorem 1. The code C n,q,m,) has parameters [q m 1)/),q m 1)/),d], where d 4. Proof: C n,q,m,) is the even-like subcode of C n,q,m,) with dimension q m 1)/). The minimum distance d 4, where the lower bound follows from the BCH bound and the upper bound follows from the sphere packing bound. Note that the parity-check matrix is H = H, where H =1,β,...,β n 1 ) is the parity-check matrix of the code C n,q,m,) in Theorem 11. We claim that every two columns of H are linearly independent over GFq). For any 1 i< j n, assume that the i-th and j-th columns are linearly dependent over GFq). Then we must have β i 1 = β j 1, that is, β j i = 1. This is impossible since β is a primitive n-th root of unity and 1 j i n 1. Hence the claim is proved. Therefore the minimum distance d of C n,q,m,) is strictly larger than two and we have d 4. For the narrow-sense BCH codes with designed distance, the parameters can be determined in some cases and are described as follows. Theorem 1. Let q. The code C n,q,m,) has parameters [q m 1)/),q m 1)/) m,d], where d can be determined in the following cases: 1). If i) q 1 mod and m, or ii) q mod and m, then d =. ). If iii) q 1 mod 4 and 4 m, or iv) q mod 4 and m,

8 then d 4. ). If q= and m, then d = 4. Proof: The dimension of C n,q,m,) follows from the fact that C 1 = C =m. By the BCH bound, the minimum distance d is at least three. We first prove d = when i) or ii) holds. Since δ=, by Lemma 9, it suffices to find a polynomial of the form σz)= i=0 σ iz i such that all roots of σz) belong to the cyclic group β and σ 1 = σ = 0. Notice that both i) and ii) imply that qm 1. Taking X 1 = 1,X = β q ),X = β qm 1) ), we find that X 1,X,X β are distinct and satisfy { X 1 + X + X = 0, X 1 X + X 1 X + X X = 0, Thus the desired polynomial can be chosen as σz)= i=1 1 X iz). Hence, there exists a codeword of weight three and we have d =. Next, we prove d 4 when iii) or iv) holds. By Lemma 9, it suffices to find a polynomial of the form σ z)= 4 i=0 σ i zi such that all roots of σ z) belong to the cyclic group β and σ 1 = σ = 0. Notice that both iii) and iv) imply that 4 qm 1 and 1 β. Thus taking X 1 = 1,X = 1,X = β q 4),X 4 = β qm 1 4), we find that X 1,X,X,X 4 β are distinct and satisfy { X 1 + X + X + X 4 = 0, X 1 X + X 1 X + X 1 X 4 + X X + X X 4 + X X 4 = 0, Thus the desired polynomial can be chosen as σ z)= 4 i=1 1 X i z). Hence, there exists a codeword of weight four and we have d 4. As for ), if q=, we have d 4 by the BCH bound. On the other hand, when q= and m, we have shown there is a codeword of weight four. Therefore we have d = 4. We make a remark here. Theoretically, the sphere packing bound gives the following restriction on the parameters of C n,q,m,) : d 1 ) n ) i q m. i i=0 Suppose m and d 7, then the above inequality leads to q m + qm 1)q m q) After a direct simplification, we have + qm 1)q m q)q m q+1) 6 q q m q + q 4 0. q m, When m, this inequality does not hold, except for q,m)=,). In addition, by Theorem 1, the code has minimum distance four when q,m)=,). Hence, for q and m, we have shown C n,q,m,) has minimum distance d 6. Together with the BCH bound, we obtain the following restriction on the minimum distance d of C n,q,m,), where m : { 4 d 6 if q=, d 6 if q>. On the other hand, Magma examples for m 6 and q 5) suggest that d = 4 when q= and d 4 when q>. These experimental results have been partially explained in Theorem 1. However, we are not sure if the method employing locator polynomials can be applied to the remaining cases, since this approach can only find the codeword whose components are either 0 or 1.

9 9 Example 1. The code C n,q,m,) has parameters [40,,4] when q,m) =,4), [165,15,] when q,m) = 4,6), and [156,14,] when q,m) = 5,4). The ternary code with parameters [40,,4] is an optimal cyclic code according to [1, p. 1]. IV. THE NARROW-SENSE BCH CODES WITH SMALL DIMENSIONS In this section, we study narrow-sense BCH codes with small dimensions. Our task is to find the first few largest coset leaders modulo n=q m 1)/). By Proposition 4, the Bose distance of a narrowsense BCH code must be a coset leader. The knowledge of these coset leaders provides information on the Bose distance and dimension of narrow-sense BCH codes whose zeroes include all roots of x n 1 except those corresponding to the first few largest coset leaders. We denote the first and the second largest coset leader modulo n by δ 1 and δ, respectively. It seems a hard problem to determine δ 1 and δ for all q>. For the rest of this section, we only deal with the case q=. A. The two coset leaders δ 1 and δ Lemma 14. Let q= and m. The first largest coset leader modulo n=q m 1)/) is and The second largest coset leader modulo n is and C δ =m. δ 1 = q 1 q )/ 1 { m if m is odd, C δ1 = if m is even. m δ = q 1 q m+1)/ 1 Proof: When m, the desired conclusions can be verified directly. Below, we consider the case that m 4. Suppose 0<δ<n is a coset leader of the form δ=a,a m,...,a 0 ). Since δ<n, we have 0 a 1. We first observe that a = 0. Otherwise, assume a = 1. Then, there is a component a i such that a i = 0. We can take a cyclic shift q j δ for some j see )) so that a i = 0 becomes the first component. This implies q j δ<δ, which is a contradiction to the assumption that δ is a coset leader. Next we assume that the coset leader δ is of the form δ=0,,a m,...,a 0 ). By the same argument as before, 00 and 01 cannot appear in the sequence δ. Moreover, if 1 appears, by taking a suitable cyclic shift we have q j δ=1,,...,0,,...) for some j. It is easy to see that 0<q j δ n=1,,...,0,,...) 1,1,...,1)=0,b,...,b 0 )=v, where 0 b 1, hence v<δ, a contradiction to the assumption that δ is a coset leader. So 1 does not appear in δ. Thirdly, let δ=0,,...,,1,...,1), 4) }{{}}{{} u v where u,v 1 and u+v+1=m. We can check that the sequences corresponding to q i δ mod n are given

10 10 by q i δ=1,...,1,0,,...,,1,...,1), 1 i u, }{{}}{{}}{{} u i v+1 i 1 q u+1+i δ=1,...,1,0,,...,,1,...,1), 0 i v 1. }{{}}{{}}{{} v i u i Therefore, the δ of the form 4) is a coset leader modulo n if and only if u v+1, that is, u m. Finally, let δ be a coset leader of the form δ=0,,a m,...,a 0 ) but not of the form 4). From what we have proved, δ must be of the form δ=0,,...,,1,...,1,0,,...,,1,...,1,,0,,...,,1,...,1), 5) }{{}}{{}}{{}}{{}}{{}}{{} u 1 v 1 u v u t v t where t,u i 1,v i 0 for each i and u 1 u i for each i t. In particular, we have u 1 u and u 1 + u + m, which implies that u 1 m 1. From the argument above, we conclude that the largest two coset leaders δ 1 and δ are given as below: δ 1 =0,,...,,1,...,1), δ }{{}}{{} =0,,...,,1,...,1), if m 5 is odd; }{{}}{{} m m+1 δ 1 =0,,...,,1,...,1), δ }{{}}{{} =0,,...,,1,...,1), if m 4 is even. }{{}}{{} m m 1 m 1 m Consequently, noting q=, we have m q i + i= δ 1 = m i=0 qi if m 5 is odd m i= m 1 qi + m i=0 qi if m 4 is even q q + q 1 if m 5 is odd = q q m 1 + q m 1 1 if m 4 is even = q 1 q )/ 1. Since q m δ 1 δ 1 mod n) when m is even, we have { m if m is odd, C δ1 = if m is even. As for δ we have m

11 11 m i= δ = m+1 m i= m q q m+1 = q q m + q m 1 q i + i=0 qi if m 5 is odd q i + m 1 i=0 qi if m 4 is even + q m+1 1 = q 1 q m+1)/ 1. if m 5 is odd if m 4 is even It can also be shown that C δ =m. This completes the proof of Lemma 14. B. Some other coset leaders δ i Let δ i denote the i-th largest coset leader modulo n =q m 1)/). In this subsection, we point out that some of the coset leaders δ i can also be determined in the case q=. Let δ be a coset leader, then according to the proof of Lemma 14, δ must have the form 4) or 5). Suppose δ has the form 5), namely, δ=0,,...,,1,...,1,...,0,,...,,1,...,1), }{{}}{{}}{{}}{{} u 1 v 1 u t v t where t, u i 1,v i 0 for each 1 i t and t i=1 u i+v i +1)=m. Since δ is a coset leader, we must have u 1 u i for each i t. Moreover, we have qδ=1,...,1,0,,...,,1,...,1,...,0,,...,), }{{}}{{}}{{}}{{} u 1 1 v 1 +1 u 1 v t +1 q u 1 δ=0,,...,,1,...,1,...,0,,...,,1,...,1). }{{}}{{}}{{}}{{} v 1 +1 u 1 v t +1 u 1 1 By the last equation, δ is a coset leader only if u 1 v i +1 for each 1 i t. Since t i=1 u i+v i +1)=m, we have u 1 +v 1 +u +v + m, which implies that u 1 m 4. This gives a strong restriction on a coset leader of the form 5). On the other hand, for 1 i, define δ i to be the following coset leaders of the form 4): δ i =0,,...,, 1,...,1 ). }{{}}{{} m+1 i m+1 +i Since u 1 m m+1 4, when i m 4, δ i is greater than any coset leader of the form 5). Thus, for 1 i m 4, δ i must be of the form 4). Indeed, we have δ i = δ i when 1 i m 4. Therefore δ i = q 1 q m +i 1, 1 i m 4. We can see the condition q= plays an essential role in the proof. For the case q>, we do not have a similar result. C. The ternary codes C n,,m,δ1 ) and C n,,m,δ1 ) In this subsection, we study the ternary codes C n,,m,δ1 ) and C n,,m,δ1 ). We recall the following trace representations of cyclic codes, which is a direct consequence of Delsarte s Theorem [11]. Note that the

12 1 length n in the following Proposition is not necessarily of the form qm 1. Proposition 15. Let q be a prime power and m = ord n q). Let γ be an n-th primitive root of unity in GFq m ) and C be a cyclic code of length n over GFq). Suppose C has s nonzeroes and let γ i 1,γ i,...,γ i s be the s roots of its parity-check polynomial which are not conjugate with each other. Denote the size of the q-cyclotomic coset C i j to be m j, 1 j s. Then C has the following trace representation where C ={ca 1,a,...,a s ) a j GFq m j ),1 j s}, ca 1,a,...,a s )= s j=1 Tr qm j q a j γ li j )) n 1 l=0. Given a codeword c, we use wc) to denote the Hamming weight of c. For the ternary code C n,,m,δ1 ), we have the following theorem. Theorem 16. Let m. Then the ternary code C n,,m,δ1 ) has parameters [ m 1, k, δ 1 ], where { m+1 if m is odd, k= m+ if m is even. In addition, C n,,m,δ1 ) is a three-weight code if m 4 is even, and a four-weight code if m is odd. The weight distribution of C n,,m,δ1 ) is listed in Table I and Table II. Proof: The conclusion on the dimension of the code follows from Lemma 14. As for the weight distribution, we consider only the case that m is odd. The case that m 4 is even can be treated in the same way and we omit the details. Note that C n,,m,δ1 ) has two nonzeroes. More precisely, 1 and β δ 1 are two non-conjugate roots of its parity-check polynomial. Since by Proposition 15, we have where m 1 δ 1 = m = +, C n,,m,δ1 ) ={c 1 a,b) : a GF m ), b GF)}, c 1 a,b)= Tr m aβ + ) j ) + b ) = Tr m aα + ) j + b ) n 1 ) n 1.

13 1 Here α is a primitive element of GF m ) and β=α. Note that gcd +, m 1)=gcd +, + 1) = gcd + 1, m+1 + 1) = gcd + 1,) =. Thus the code C n,,m,δ1 ) is permutation equivalent to the following code { c a,b)= Tr m aα j ) } n 1 + b) : a GFm ), b GF). 6) Because of the simple expression of the code 6) and because the weight distribution problem has been studied extensively for many families of cyclic codes in recent years see for example [16] for a recent update and references therein), the weight distribution of the code in 6) may be known, but we have not found it in the literature. For the sake of completeness and the illustration of ideas and methods which will be used later, we include a detailed computation below. First, if a=0 and 0 b GF), it is easy to see that the Hamming weight wc a,b)) of the codeword c a,b) takes the value m 1 twice. Second, if 0 a GF m ) and b GF), we have By ) we obtain wc a,b))=n {0 j n 1 Tr m aα j )+b=0} wc a,b))= n 1 6 n 1 1 = n x GF) = n 1 ζ bx x GF) = n 1 6 ζ xtrm aα j )+b) n 1 ζ Trm axα j ) ζ bx ζ Trm axy ). x GF) y GF m ) ζ bx ηax)gη) 1) x GF) = n 1 6 ηa)gη) ζ b + η 1)ζ b ) ζ b + ζ b where η is the quadratic character of GF m ) and Gη) is the quadratic Gauss sum over GF m ). Since m is odd, η 1)= 1. Using the values of Gη) from Lemma 7, we find that if a 0 and b=0, wc a,b)) takes the value for m 1 times, and if a 0 and b 0, the weight wc a,b)) takes each of the values 1+ 1)m+1)/ )/ and 1+ 1))/ )/ for m 1 times. Therefore, we obtain the weight distribution for the case that m is odd, which is recorded in Table II. Example. Let q,m)=,4). Then the code C n,q,m,δ1 ) of Theorem 16 has parameters [40,,5], and weight enumerator 1+16z 5 + z 0 + z 40. This code is the best cyclic code according to [1, p. 05]. Example. Let q,m)=,5). Then the code C n,q,m,δ1 ) of Theorem 16 has parameters [11,6,76], and weight enumerator 1+4z z 1 + 4z 5 + z 11. ),

14 14 TABLE I THE WEIGHT DISTRIBUTION OF C n,q,m,δ1 ) WHEN m 4 IS EVEN Weight Frequency 0 1 m/ 1 +1 m/ 1) + m/ 1 m/ 1 m 1 TABLE II THE WEIGHT DISTRIBUTION OF C n,q,m,δ1 ) WHEN m IS ODD Weight Frequency 0 1 m ))/ )/ m )m+1)/ )/ m 1 m 1 Theorem 17. Let m. Then the ternary code C n,,m,δ1 ) has parameters [ m ] 1, k, d, where and d = { m if m is odd, k= if m is even, In addition, C n,,m,δ1 ) has only one nonzero weight. m { if m is odd, + m 1 if m is even. Proof: The conclusion on the dimension of the code follows from Lemma 14. Note that C n,,m,δ1 ) has one nonzero and β δ 1 is a root of its parity-check polynomial. By using Proposition 15, we have C n,,m,δ1 ) ={c 1 a) : a GF m )},

15 15 where c 1 a)= = Tr m )) aβ + n 1 ) j Tr m aα + ) j )) n 1 We consider only the case that m is odd. The case that m 4 is even can be treated in the same way. In this case, as was shown in the proof of Theorem 16, since gcd +, m 1)=, C n,,m,δ1 ) is permutation equivalent to the following code { c a)= Tr m aα j )) } n 1, a GFm ). 7) Again the weight distribution of the code of 7) may be known, but we have not found it in the literature. We provide details of the computation as below. Clearly, c 0) is the zero codeword. If 0 a GF m ), using ), we have. wc a))=n {0 j n 1 Tr m aα j )=0} n 1 1 = n x GF) = n 1 x GF) = n 1 6 ζ xtrm aα j )) n 1 ζ Trm axα j ) ζ Trm axy ) x GF) y GF m ) = n 1 6 x GF) ηax)gη) 1) = 1 6 Gη) x GF) ηax), where η is the quadratic character of GF m ) and Gη) is the quadratic Gauss sum over GF m ). Since η1)=1 and η 1)= 1, we have x GF) ηax)=0 for each a 0. Therefore, wc a))= for each a 0. Namely, C n,,m,δ1 ) has only one nonzero weight. We remark that, as it could be easily verified, the code C n,,m,δ1 ) of Theorem 17 meets the Griesmer bound for linear codes, and hence is optimal. D. The ternary codes C n,,m,δ ) and C n,,m,δ ) The determination of the weight distributions of the ternary codes C n,,m,δ ) and C n,,m,δ ) is more complicated and depends heavily on the theory of quadratic forms over finite fields see []). We first do some preparations. For reasons which will be explained later, let m be an odd integer. For a,b GF m ), define the quadratic form Q a,b x)=tr m Let r a,b be the rank of the quadratic form Q a,b. ) ax +1 + bx m +1.

16 16 Lemma 1. Let m be odd, a,b GF m ) and a,b) 0,0). The quadratic form ) Q a,b x)=tr m ax +1 + bx m +1 has rank r a,b {m,,m,m }. Proof: Let B a,b x,y) be the symmetric bilinear form associated with the quadratic form Q a,b, that is, Recall that the equation B a,b x,y)=q a,b x+y) Q a,b x) Q a,b y) b m+ x m+ = Tr m b m+ x m+ + a m+1 x m+1 + ax + bx m + a m+1 x m+1 + ax + bx m = 0 ) ) y. has r solutions x GF m ) if and only if the rank of Q a,b equals m r see [, p. 1], for instance). Note that the number of solutions of the above equation equals the number of solutions of the following one: b m+ x + a m+1 x + ax + bx=0. This has at most 7 solutions, thus r and r a,b {m,,m,m }. For a,b GF m ), define Ta,b)= x GF m ) ζ Trm ax +1 +bx m +1 ). Denote by η 0 the quadratic character over GF). Then, by Lemma, we have Sa,b) := y GF) Tya,yb) = y GF) η 0 y r a,b )Ta,b) = Ta,b)1+ 1) r a,b ) ) We will need the following lemma concerning power moment identities. Lemma 19. For the Sa,b) defined above, we have the following: i) a,b GF m ) Sa,b)= m. ii) a,b GF m ) Sa,b) = 4 m. iii) a,b GF m ) Sa,b) = m 4 m. iv) a,b GF m ) Ta,b) = m. Proof: i) The identity is trivially true. ii) Define N to be the number of pairs u,v) GF) GF) and x,y) GF m ) GF m ), which satisfy the following two equations: ux +1 + vy +1 = 0, ux m +1 + vy m +1 = 0. It is easy to see that a,b GF m ) Sa,b) = m N. Thus, it suffices to determine N. If one of x,y is zero, then the other must also be zero. When x=y=0, we have four choices of the

17 17 pair u,v). When x 0 and y 0, the system above is equivalent to x y ) +1 = u v, m ) +1 = u v. As shown in the proof of Theorem 16, we have gcd + 1, m 1)=. Similarly, gcd m + 1, m 1)=gcd m + 1, m+ + 1) x y = gcd m + 1, 1) gcd m 1+, 1) if m 0 mod = gcd m +4, 1) if m 1 mod gcd m 9+10, 1) if m mod gcd, 1) if m 0 mod = gcd4, 1) if m 1 mod gcd10, 1) if m mod = Thus, u v GF) must be a square. Namely, u v = 1. Thus, we have two choices for the pair u,v). Meanwhile, y x =±1 are precisely all solutions to the following system of equations: x y ) +1 = 1, m ) +1 = 1. x y Thus, we have m 1) choices for the pair x,y). To sum up, we have 4 m 1) choices for the pairs u,v) and x,y) when x 0 and y 0. In total, we have N = 4+4 m 1)=4 m. iii) Define N to be the number of triples u,v,w) GF) GF) GF) and x,y,z) GF m ) GF m ) GF m ), which satisfy the following two equations: ux +1 + vy +1 + wz +1 = 0, ux m +1 + vy m +1 + wz m +1 = 0. It is easy to see that a,b GF m ) Sa,b) = m N. Thus, it suffices to determine N. If two of x,y,z are zero, then the third one must also be zero. When x=y=z=0, we have eight choices of the triple u,v,w). When exactly one of x,y,z equals 0, we can use the result of ii). For instance, if x=0, then u {1,} and the system degenerates to vy +1 + wz +1 = 0, vy m +1 + wz m +1 = 0. By ii), we have 4 m 1) choices for the pairs v,w) and y,z). Thus, we have m 1) choices for the triples u,v,w) and 0,y,z). In total, when exactly one of x,y,z equals 0, we have 4 m 1) choices. When x,y,z are all nonzero, we have u x ) + v y ) + w=0, z z 9) u x m ) + v y m ) + w=0. z z 10)

18 1 Raising to the m+ -th power both sides of 9) and 10), we obtain Taking the difference between 11) and 1), we have u x m+ ) + v y m+ ) + w=0, z z 11) u x m+ ) + v y m+ ) + w=0. z z 1) u x m+ ) + x m+ ) +1 )+v y m+ ) + y m+ ) +1 )=0. z z z z Dividing y m+ z ) +1 on both sides of the above equation, we can easily obtain Raising to the third power both sides of 10), we obtain Taking the difference between 14) and 9), we have x m+ ) +1 = vy z ) y ux z ). 1) u x ) + + v y ) + + w=0. 14) z z u x ) + x ) +1 )+v y ) + y ) +1 )=0 z z z z Dividing y z ) +1 on both sides of the above equation, we can easily obtain x ) +1 = vy z ) y ux z ). 15) By 1) and 15), we have x m+ y ) +1 = x y ) +1, which is equivalent to x y ) = 1. Since gcd, m 1)=, we have y x) = 1. By the symmetry of x,y,z, we also conclude that x z ) = 1 and y z ) = 1. Hence, the original system degenerates to u+v+w=0, u,v,w GF). It is easy to verify that there are two choices of the tripleu,v,w) and 4 m 1) choices of the triplex,y,z). In total, there are m 1) choices when x,y,z are all nonzero. Hence, N = +4 m 1)+ m 1)= m 4. iv) The proof is very similar to that of ii), and thus omitted here. For j {0,}, define {a,b) GF n j = m ) GF m ) : Ta,b)=± m+ j }. 1 For j {1,} and ε {1, 1}, define {a,b) } n ε, j = GF m ) GF m ) : Ta,b)=ε m+ j. Under the assumption n 1, = 0, which will be verified later see the proof of Theorem 1 below), we obtain the value distribution of Ta,b) and Sa,b). Lemma 0. Let m be an odd integer. Assume that n 1, = 0. 1) The value distribution of Ta,b) is as follows:

19 19 Rank r a,b Value Ta,b) Multiplicity m m 1 m 1) m 9 +9) m m 1 m 1) m 9 +9) m+1 m+1 m m m +1 1 m +1 1 ) The value distribution of Sa, b) is as follows: + ) m 1) ) m 1) m 1) 1) m 1) 1) 0 m 1 Rank r a,b Value Sa,b) Multiplicity m,m 0 m + 1) m 1) m+1 m+1 + ) m 1) ) m 1) 0 m 1 Proof: By ), we have Sa,b)= { 0 if r a,b {m,m }, Ta,b) if r a,b {,m }. Therefore, the value distribution of Sa,b) easily follows from that of Ta,b). Note that T0,0)= m and S0,0)= m. Below, we are going to determine the value distribution of Ta,b), employing the four moment identities in Lemma 19. Indeed, the moment identities lead to the following four equations: a,b GF m ) a,b GF m ) a,b GF m ) a,b GF m ) Sa,b)= S0,0)+ r a,b {,m } Ta,b) = m + m+1 n1,1 n 1,1 )+ m+ n1, n 1, ))= m. Sa,b) = S0,0) + r a,b {,m } Ta,b)) = 4 m + 4 m+1 n 1,1 + n 1,1 )+ m+ n 1, + n 1, ))=4 m. Sa,b) = S0,0) + r a,b {,m } Ta,b)) = m + m+1) n 1,1 n 1,1 )+ m+) n 1, n 1, ))= m 4 m. Ta,b) = T0,0) + Ta,b) a,b) 0,0) = m m n 0 + m+1 n 1,1 + n 1,1 ) m+ n + m+ n 1, + n 1, )= m.

20 0 Simplifying the above four equations leads to Moreover, by definition, we have n 1,1 n 1,1 + n 1, n 1, )= m 1), n 1,1 + n 1,1 + 9n 1, + n 1, )= m 1), n 1,1 n 1,1 + 7n 1, n 1, )= m 1), n 0 + n 1,1 + n 1,1 ) 9n + 7n 1, + n 1, )=0. n 0 + n + n 1,1 + n 1,1 + n 1, + n 1, = m 1. Together with n 1, = 0, we already have six linear equations with respect to n 0,n,n 1,1,n 1,1,n 1, and n 1,, from which the value distribution of Ta,b) easily follows. Theorem 1. Let m. Then the ternary code C n,,m,δ ) has parameters [ m ] 1, k, d, where { m if m is odd, k= if m is even, m and d =. Moreover, the weight distribution is presented in Table III when m is even and in Table IV when m is odd. Proof: The conclusion on the dimension of the code follows from Lemma 14. C n,,m,δ ) has two nonzeroes and β δ 1, β δ are two non-conjugate roots of its parity-check polynomial. Note that Define c 1 a,b)= = Tr m Tr m Tr m Tr m m 1 m 1 aβ + δ 1 = m = +, δ = m = + m+1. ) j aβ m + 1 ) j + bβ + ) + Tr m m+1 ) j 1 m+1 1 )) n 1 bβ m )) n 1 + ) j )) n 1 aα + ) j + bα + m+1 ) j aα + m ) 1 ) j + Tr m bα + m ) j )) n 1 if m is odd if m 4 is even if m is odd if m 4 is even where a,b GF m ) for m being odd and a GF m ), b GF m ) for m being even. By Proposition 15,

21 1 we have when m is odd and C n,,m,δ ) ={c 1 a,b) : a,b GF m )} C n,,m,δ ) ={c 1 a,b) : a GF m ),b GF m )} when m is even. When m is even, we have Tr m ) aα + m 1 ) j ) bα + m ) j Tr m = Tr m = Tr m Since x x is a permutation of GF m ) and x x m same weight distribution with the following code: { ) Tr m aα m +1) j + Tr m bα m 1 +1) j ) a α m +1) j, b m α m 1 +1) j ). is a permutation of GF m ), C n,,m,δ ) has the )) n 1, a GF m ),b GF m ) }. 16) We remark that the weight distribution of the following cyclic code has been studied in [4, Theorem ]: { ) )) } Tr m aα m +1) j + Tr m bα m n ) j, a GF m ),b GF m ). 17) Note that there is a one-to-one correspondence between the codewords of 16) and 17). Indeed, given a codeword of 16), the concatenation with its copy produces a codeword of 17). Hence, the weight of a codeword in code 16) is half of its corresponding codeword in code 17). Therefore, the weight distribution of code 16) easily follows from that of code 17) [4, Theorem ]. Thus, the weight distribution of C n,,m,δ ) is obtained and presented in Table III. It is also known that the minimum distance is d = m 1. When m is odd, we need some extra work. Note that ) Tr m Tr m aα + ) j bα + m+1 ) j ) = Tr m = Tr m a m+1 b α ) +1) j α m +1) j Since x x m+1 and x x are both permutations of GF m ), C n,,m,δ ) has the same weight distribution with the following code where c a,b)= {c a,b) : a,b GF m )}., ). )) Tr m +1) aα j m n 1 +1) j + bα.

22 TABLE III THE WEIGHT DISTRIBUTION OF C n,,m,δ ) WHEN m 4 IS EVEN Weight Frequency 0 1 m 1 m 1) m +1) m 1 m 1) + m 1 m 1) +1) 4 + m m 1 1) m 1) Note that wc a,b))= n {0 j n 1 Tr m +1) aα j m +1) + bα j )=0} = n = n 1 6 n 1 1 y GF) = n 1 6 n 1 y GF) ζ ytrm ζ ytrm aα +1) j m +bα +1) j )) aα +1) j m +bα +1) j )) ζ ytrm ax +1 +bx m +1 )) y GF) x GF m ) = n = m 1 ζ Trm yax +1 +ybx m +1 ) y GF) x GF m ) + 1 m = 1 6 y GF) Tya,yb) ζ Trm yax +1 +ybx m +1 ) y GF) x GF m ) = 1 Sa,b) 1) 6 Hence, we obtain the weight distribution of C n,,m,δ ) see Table IV) directly from the value distribution of Sa,b) in Lemma 0, provided that n 1, = 0. We finally prove that n 1, = 0. By 1) and Lemma 0, n 1, is the frequency of codewords of C n,q,m,δ ) which have the weight m+1. However, C n,,m,δ ) is a subcode of C n,,m,δ ), whose minimum distance satisfies d δ = 1 m+1 1 > m+1. Therefore, there must be no codeword of weight m+1, which forces n 1, = 0. This concludes the proof of Theorem 1. Example 4. Let q,m)=,4). Then the code C n,q,m,δ ) of Theorem 1 has parameters [40,6,4], and weight enumerator 1+00z 4 +40z 7 +16z 0 +0z 6. This is the best cyclic code and optimal according to [1, p. 05].

23 TABLE IV THE WEIGHT DISTRIBUTION OF C n,,m,δ ) WHEN m IS ODD Weight Frequency ) m 1) m + 1) m 1) + ) m 1) Example 5. Let q,m)=,5). Then the code C n,q,m,δ ) of Theorem 1 has parameters [11,10,7], and weight enumerator z z z 90. This code has the same parameters as the best ternary linear code known in the Database. Theorem. Let m. Then the ternary code C n,,m,δ ) has parameters [ m 1, k, δ ], where { m+1 if m is odd, k= m+ if m is even. In addition, the weight distribution is presented in Table V when m is even and in Table VI when m is odd. Proof: The conclusion on the dimension of the code follows from Lemma 14. Note that C n,,m,δ ) has three nonzeroes and 1, β δ 1, β δ are three roots of its parity-check polynomial, such that every two of them are non-conjugate. Define c 1 a,b,c)= = Tr m Tr m Tr m Tr m aβ + ) j aβ m + 1 ) j + bβ + ) + Tr m m+1 ) j ) + c bβ m + ) j ) n 1 ) + c ) ) n 1 aα + ) j + bα + m+1 ) j + c aα + m ) 1 ) j + Tr m bα + m ) ) j + c ) n 1 ) n 1 if m is odd if m 4 is even if m is odd if m 4 is even where a,b GF m ), c GF) for m being odd and a GF m ), b GF m ), c GF) for m being even. Similar to the proof of Theorem 1, by using Proposition 15, we have when m is odd and C n,,m,δ ) ={c 1 a,b,c) : a,b GF m ),c GF)} C n,,m,δ ) ={c 1 a,b,c) : a GF m ),b GF m ),c GF)}

24 4 when m is even. When m is even, with the same argument as in the proof of Theorem 1, we know that C n,q,m,δ ) has the same weight distribution with the following code { } c a,b,c) : a GF m ),b GF m ),c GF), where c a,b,c)= ) ) ) Tr m aα m +1) j + Tr m bα m n ) j + c. Recall that η 0 is the quadratic character of GF). With the help of Lemma, we have wc a,b,c))=n {0 j n 1 Tr m aα m +1) j )+Tr m n 1 1 = n y GF) = n 1 6 n 1 = n 1 6 y GF) m ζ ytr aα m +1) j )+Tr m m bα 1 +1) j )+c) m ζ ytr = n+ 1 6 ζ yc 1 6 y GF) = m 1 aα m +1) j )+Tr m m bα 1 +1) j )+c) m ζ ytr ax m +1 )+Tr m m bx 1 +1 )+c) y GF) x GF m ) m ζ Tr yax m +1 )+Tr m y GF) x GF m ) + 1 δ 0,c m bα m 1 +1) j )+c=0} ybx m 1 +1 )+yc m ζ Tr yax m +1 )+Tr m m ybx 1 +1 )+yc y GF) x GF m ) where = + 1 δ 0,c 1) 1 6 = + 1 δ 0,c 1) 1 6 ζ yc Uya,yb) y GF) m ζ yc ζ Tr yax m +1 )+Tr m m ybx 1 +1 ) y GF) x GF m ) = + 1 δ 0,c 1) 1 6 ζ yc η 0y) r a,bua,b) y GF) Ua,b)= δ 0,c = x GF m ) { 1 if c=0, 0 if c 0, ζ Trm ax m +1 +bx m 1 +1 ), and r a,b is the rank of the quadratic form m Trm ax +1 +bx m 1 +1 ). Thanks to [4, Theorem 1], the value distribution of Ua,b) is already known which is presented in the following table:

25 5 Rank r a,b Value Ua,b) Multiplicity m m m m m+1 i m+1 i m 1) m +1) m 1) +1) 4 m 1 m 1) m 1 m 1) m m +1 m 1 1) m 1) 0 m 1 It is easy to see that the weight distribution of C n,q,m,δ ) see Table V) follows directly from the value distribution of Ua,b) when m is even. For example, according to the table above, when r a,b = m, Ua,b) takes the value m for m 1) m +1) times. Thus, if c=0, wc a,b,c)) takes the value m 1 for m 1) m ) +1) times, and if c = 1 or, wc a,b,c)) takes the value + 1 m 1 1 for m 1) m +1) 4 times. When m is odd, with the same argument as in the proof of Theorem 1, we know that C n,q,m,δ ) shares the same weight distribution with the following code where c a,b,c)= {c a,b,c) : a,b GF m ),c GF)}, ) ) Tr m +1) aα j m n 1 +1) j + bα + c. Very similar to the above case of wc a,b,c)), we can show that wc a,b,c))= + 1 δ 0,c 1) 1 6 ζ yc η 0y) r a,b Ta,b). y GF) Employing the value distribution of Ta,b) in Lemma 0, we obtain the weight distribution of C n,,m,δ ) see Table VI) directly when m is odd. This completes the proof of Theorem. Example 6. Let q,m)=,4). Then the code C n,q,m,δ ) of Theorem has parameters [40,7,], and weight enumerator 1+0z + 00z 4 + 6z z z + 16z z 1 + 0z 6 + z 40. According to [1, p. 05], this is the best ternary cyclic code, and has the same parameters as the best ternary linear code in the Database. Note that for each ternary linear code with parameters [40,7,d], we have d. Example 7. Let q,m)=,5). Then the code C n,q,m,δ ) of Theorem has parameters [11,11,67], and weight enumerator 1+40z z z z z z z 94 + z 11. The best ternary linear code known in the Database has parameters [11,11,6], which is not known to be cyclic.

26 6 TABLE V THE WEIGHT DISTRIBUTION OF C n,,m,δ ) WHEN m 4 IS EVEN Weight Frequency 0 1 m m 1 m m 1 m m m 1 1) m 1) 4 m 1) m +1) m 1) +1) m m 1 m 1) m + m 1 1 m + m 1 m 1) m +1) 4 m 1) +1) 4 m + m 1 m 1 m 1) m + m m 1 1) m 1) m 1 TABLE VI THE WEIGHT DISTRIBUTION OF C n,,m,δ ) WHEN m IS ODD Weight Frequency 0 1 m m+1 1 m m 1 1) m 1) + ) m 1) m +9) m 1) m m + 1) m 1) m + 1 m + m + m+1 1 m + +9) m 1) ) m 1) 1) m 1) m 1

27 7 V. SOME NARROW-SENSE BCH CODES WITH SPECIAL DESIGNED DISTANCES In this section, for general prime power q, we focus on narrow-sense BCH codes of length n= qm 1 with special designed distances. As stated in Section II, some information about coset leaders modulo q m 1 can be obtained via the NDS decomposition. Employing Proposition 4, we can obtain the Bose distance if the designed distance is of certain special form. Theorem. Let δ n be an integer. Suppose Eδ)= V 1 V...V r. i) Suppose δ has only 0 and 1 as components. If V 1 = V = = V r or V 1 = V = = V j < V k for some j and all k such that j< k r, then C n,q,m,δ) has Bose distance d B = δ. In particular, if Eδ)= V 1, then C n,q,m,δ) has Bose distance d B = δ. ii) Suppose V 1 has length l and has components either 0 or 1. Let V 1 > V and m = al+ b, where 0 b l 1. 1) If b=0, then C n,q,m,δ) has Bose distance d B = E 1 V 1 V 1...V 1 ). }{{} a ) If 1 b l 1, then C n,q,m,δ) has Bose distance d B E 1 V 1 V 1...V 1 ST b V 1 ))). }{{} a In particular, if the last component of ST b V 1 )) is 1, then the equality holds. Proof: These results are immediate consequences of Proposition 4, Lemma 5 and Lemma 6. The theorem above is very powerful in determining the Bose distance of C n,q,m,δ) for some special δ. In the following, we give several examples. Example. For q=, m=6 and δ=110, consider the code C 64,,6,110). Note that δ=0,1,1,0,0,)=v 1 V, where V 1 =0,1,1) and V =0,0,). Since V 1 > V, by ii) of Theorem, the Bose distance d B = E 1 V 1 V 1 )=E 1 0,1,1,0,1,1)= 11. Indeed, the smallest coset leader no less than δ=110 is just 11, where C 11 ={11,0,6}. Example 9. For q=, m=5 and δ=9, consider the code C 11,,5,9). Note that δ=0,1,0,0,)=v 1 V, where V 1 =0,1) and V =0,0,). Since V 1 > V, by ii) of Theorem, the Bose distance d B = E 1 V 1 V 1 ST 1 V 1 )))=E 1 0,1,0,1,1)= 1. Indeed, the smallest coset leader no less than δ=9 is just 1, where C 1 ={1,7,91,9,111}. Example 10. For q=7, m=5 and δ=9, consider the code C 01,7,5,9). Note that δ=0,1,1,0,1)=v 1 V, where V 1 =0,1,1) and V =0,1). Since V 1 > V, by ii) of Theorem, the Bose distance d B E 1 V 1 ST V 1 )))=E 1 0,1,1,0,)= 94. Indeed, the smallest coset leader no less than δ=9 is just 94, where C 94 ={94,694,057,500,75}.

28 Hence, we have the Bose distance d B = 94. Below, we consider two special classes of designed distances. First, we consider BCH codes with designed distances qi 1, where 1 i. Theorem 4. For 1 i, C n,q,m, q i has Bose distance d 1 ) B = qi 1. Furthermore, if 1 i m, then the code C n,q,m, q i 1 has parameters ) [ q m ] 1, qm 1 mqi 1 1), d, where d qi 1. In particular, if i m, then d = qi 1. Proof: Note that the sequence q i 1 =0,...,0,1,...,1) }{{} m i is an NDS. By iii) of Lemma 5, qi 1 is a coset leader. Then, by Proposition 4, we have d B= qi 1. When 1 i m, the conclusion on dimensions follows from Theorem. If i m, by Lemma 10, we have d = qi 1. It would be good if the following open problem can be settled. Open Problem 1. Dose the code C n,q,m,q i 1)/)) have minimum distance d =qi 1)/), where 1 i? Note that Theorem 4 gives an affirmative answer to the open problem, when i m. Next, we consider BCH codes with designed distances q i + l, where 1 i m 1 and 1 l. Theorem 5. For 1 i m 1 and 1 l, C n,q,m,q i +l) has Bose distance d B = q i +l. Furthermore, the code C n,q,m,q i +l) has parameters [ q m 1, qm 1 m q i + l 1) 1 1 ) ], d, q where d q i + l. In particular, if q i + l) n, then d = q i + l. Proof: The conclusion on the dimension follows from Theorem. For 1 i m 1 and 1 l, let δ=q i + l. To prove that the Bose distance is equal to δ, by Proposition 4, it suffices to show that δ is a coset leader. Note that δ=0,...,0,1,0,...,0,l). }{{}}{{} m i 1 i 1 We are going to show that δ is a coset leader by analyzing δ. Direct computation shows that for 1 i m, q i δ>δ. Moreover, 1,0,...,0,1,0,...,0) if l = 1, }{{}}{{} q m i 1 i 1 δ= 0,q l,...,q l,q l+ 1,q l,...,q l,q l+ 1) if l, }{{}}{{} m i 1 i which implies that q δ>δ. Consequently, δ is a coset leader modulo n. In addition, if q i + l) n, by Lemma 10, we have d = q i + l.