A Family of Reversible BCH Codes

Transcription

1 1 A Faily of Reversible BCH Codes Shuxing Li, Cunsheng Ding, and Hao Liu Abstract arxiv: v1 [cs.it] 7 Aug 016 Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algiths as well as their theetical iptance. BCH codes f a subclass of cyclic codes and are very iptant in both they and practice as they have good err-crecting capability and are widely used in counication systes, stage devices and consuer electronics. However, the diension and iniu distance of BCH codes are not known in general. The objective of this paper is to study the diension and iniu distance of a faily of BCH codes over finite fields, i.e., a class of reversible BCH codes. I. INTRODUCTION Throughout this paper, let p be a prie and let q be a power of p. An [n,k,d] linear code C over GF(q) is a k-diensional subspace of GF(q) n with iniu (Haing) distance d. A linear code C over GF(q) is called a linear code with copleentary dual (in sht, LCD) if C C ={0}, where C denotes the dual of C. An [n,k] linear code C is called cyclic if (c 0,c 1,,c n 1 ) C iplies (c n 1,c 0,c 1,,c n ) C. By identifying any vect (c 0,c 1,,c n 1 ) GF(q) n with c 0 + c 1 x+c x + +c n 1 x n 1 GF(q)[x]/(x n 1), any code C of length n over GF(q) cresponds to a subset of the quotient ring GF(q)[x]/(x n 1). A linear code C is cyclic if and only if the cresponding subset is an ideal of the ring GF(q)[x]/(x n 1). Note that every ideal of GF(q)[x]/(x n 1) is principal. Let C = g(x) be a cyclic code, where g(x) is onic and has the sallest degree aong all the eleents of C. Then g(x) is unique and called the generat polynoial, and h(x)=(x n 1)/g(x) is referred to as the parity-check polynoial of C. Let f(x) GF(q)[x] be a polynoial with degree l, then the reciprocal polynoial of f is defined to be f R (x)=x l f(x 1 ). f is called self-reciprocal if and only if f(x)= f R (x). A cyclic code C with generat polynoial g(x) is called reversible if g(x) is self-reciprocal. The reversibility iplies if then (c 0,c 1,...,c n 1 ) C, (c n 1,c n,...,c 0 ) C. It is easily seen that reversible cyclic codes are LCD codes. Let 1 be a positive integer and let n=q 1. Let α be a generat of GF(q ). F any i with 1 i q, let i (x) denote the inial polynoial of α i over GF(q). F any δ<n, define g + (q,,δ) (x)=lc( 1(x), (x),, δ 1 (x)), and g (q,,δ) (x)=lc( n δ+1(x), n δ+ (x),, n 1 (x)). S. Li is with the Departent of Matheatics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (E-ail: lsxlsxlsx1987@gail.co). C. Ding is with the Departent of Coputer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (E-ail: cding@ust.hk). H. Liu is with the Departent of Coputer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (E-ail: hliuar@connect.ust.hk).

2 where lc denotes the least coon ultiple of these inial polynoials. Let C (q,,δ) denote the cyclic code of length n with generat polynoial g + (q,,δ) (x). This code C (q,,δ) is called the narrowsense priitive BCH code with designed distance δ. The codes C (q,,δ) have been studied in series of papers, including [1], [3], [4], [5], [7], [8], [9], [11], [1], [16], [18], [5]. However, the diension and iniu distance of the codes C (q,,δ) are still open in general. F any δ<(n+)/, define g (q,,δ) (x)=lc(g (q,,δ) (x),g+ (q,,δ) (x)). Let C (q,,δ) denote the cyclic code of length n with generat polynoial g (q,,δ) (x). C (q,,δ) is a reversible cyclic code which was studied by Massey f the data stage applications [19]. F the code C (q,,δ) with length n and iniu distance d, it was shown in [4] that { d = δ if δ n, d δ+1 otherwise. Meover, if we consider the even-like subcode of C (q,,δ), naely, the code C (q,,δ) with length n and generat polynoial g (q,,δ) (x)=(x 1) g (q,,δ) (x), its iniu distance is at least δ by the BCH bound. Hence, a potential great iproveent on the iniu distance is expected by considering the even-like subcode of C (q,,δ). This intuition otivates us to study the code C (q,,δ), which is a reversible BCH code. Recently, LCD codes have found an iptant application in cryptography [6]. Meover, it was shown that asyptotically good LCD codes exist [13], [0], [3]. These are our aj otivations of studying reversible BCH codes, which are a special class of LCD codes. The objective of this paper is to study the fundaental paraeters, naely diensions and iniu distances of the reversible BCH codes C (q,,δ). II. THE DIMENSION OF THE REVERSIBLE BCH CODES C (q,,δ) Our task in this section is to study the diension of the codes C (q,,δ). It is in general a hard proble to deterine the diension of a BCH code. A. The diension of C (q,,δ) when δ is relatively sall In [9], the auths derived the diension of the narrow-sense priitive BCH code C (q,,δ) when δ is relatively sall. In this subsection, we develop siilar results f the diension of the reversible BCH code C (q,,δ). Any positive integer s with 0 s n has a unique q-ary expansion as s= i=0 s iq i, where 0 s i q 1. We use C δ to denote the q-cyclotoic coset in Z n containing δ, which is defined by C δ ={δq i od n : 0 i }. We use cl(δ) to denote the coset leader of C δ, which is the sallest integer in C δ. The q-ary expansion sequence of s= i=0 s iq i is denoted as s=(s,s,...,s 0 ). Below, we siply call the q-ary expansion sequence of s as the sequence of s, whenever this causes no confusion. The weight of s is defined to be the nuber of nonzero entries aong s and denoted by wt(s). Define the suppt of s as supp(s)={0 i s i 0}. We have the following properties regarding q-cyclotoic cosets odulo n=q 1. Lea 1. Let and n=q 1. Then we have the following.

3 3 1) When is odd, f 1 i q (+1)/, C i = C i =. F 1 i, j q (+1)/, j C i, equivalently j C i, if and only if j q l i (od n) f soe 0 l. ) When is even, C q / +1 = C q / 1 = and C i = C i = f 1 i q /, i q / +1. F 1 i, j q /, j C i, equivalently j C i, if and only if f soe 0 l. j q l i (od n) Proof: 1) The proof is essentially the sae as that of [9, Thee 3] (see also [1, Leas 8,9]), and is oitted. ) The proof is essentially the sae as that of [9, Thee 3], and is oitted. Lea. Let and let n=q 1. Then the following stateents hold. 1) When is odd, f 1 i, j q (+1)/, j C i if and only if i=(0,...,0,q 1,...,q 1,u), j=(0,...,0,q 1 u,q 1,...,q 1), 1 3 i=(0,...,0,q 1 u,q 1,...,q 1), 3 j=(0,...,0,q 1,...,q 1,u), where 0 u q 1. ) When is even and q>, f 1 i, j q /, j C i if and only if i=(0,...,0, 1,q 1,...,q 1,q ), 1 j =(0,...,0, 1,q 1,...,q 1,q ), 1 i=(0,...,0,q 1,...,q 1,q ), 1 1 j=(0,...,0, 1,q 1,...,q 1), 1 i=(0,...,0, 1,q 1,...,q 1), 1 j=(0,...,0,q 1,...,q 1,q ), 1 1 i=(0,...,0,q 1,...,q 1), 1 j =(0,...,0,q 1,...,q 1). 1 1

4 4 3) When is even and q=, f 1 i, j (/)+1, j C i if and only if i=(0,...,0, 1,...,1), j =(0,...,0, 1,...,1), i=(0,...,0, 1,...,1), j =(0,...,0, 1,...,1), i=(0,...,0, 1 1,...,1), j =(0,...,0, 1 1,...,1), i=(0,...,0, 1,0,1,...,1), j=(0,...,0, 1,...,1,0,1), i=(0,...,0, 1,...,1,0,1), j=(0,...,0, 1,0,1,...,1). Proof: 1) If j C i, then there exists an l with 0 l, such that q l i+ j 0 (od n). Hence, q l i+ j=(q 1,q 1,...,q 1). Since =wt(q l i+ j) wt(i)+wt( j) +1, we have {wt(i),wt( j)}={ } {wt(i),wt( j)}= { +1 }. If {wt(i),wt( j)} = {, +1 }, then clearly, supp(ql i) supp( j) = /0. Otherwise, if supp(q l i) supp( j) /0, there is at least one entry in q l i+ j, which is not q 1. Hence, q l i and j ust have the following two fs q l i=(q 1,...,q 1,0,...,0), +1 j =(0,...,0,q 1,...,q 1), q l i=(q 1,...,q 1,0,...,0), j =(0,...,0,q 1,...,q 1), 3, +1

5 5 If {wt(i),wt( j)}={ +1 }, then clearly, supp(ql i) supp( j) 1. If supp(q l i) supp( j) >1, then there is at least one entry in q l i+ j, which is not q 1. Hence, supp(q l i) supp( j) =1. Therefe, q l i and j ust have the following q 4 fs q l i=(q 1,...,q 1,u,0,...,0), +1 j =(0,...,0,q 1 u,q 1,...,q 1), q l i=(q ,...,q 1, 0,...,0,u), j =(0,...,0,q 1,...,q 1,q 1 u), where 1 u q. Therefe, the conclusion follows. ) If j C i, then there exists an l with 0 l, such that q l i+ j 0 (od n). Hence, q l i+ j=(q 1,q 1,...,q 1). Since =wt(q l i+ j) wt(i)+wt( j) +, we ust have { + 1}, { {wt(i),wt( j)}=, + 1}, { 1, + 1}, { }. If {wt(i),wt( j)}={ +1}, then supp(ql i) supp( j) =. Hence, q l i and j ust have the following f q l i=(q 1,...,q 1,q, 0,...,0,1), 1 +1 j=(0,...,0, 1,q 1,...,q 1,q ). 1 If {wt(i),wt( j)}={, + 1}, then supp(ql i) supp( j) =1. Hence, q l i and j ust have the following two fs q l i=(q 1,...,q 1,q,0,...,0), +1 j=(0,...,0, 1,q 1,...,q 1), 1 q l i=(q 1,...,q 1, 0,...,0,1), 1 j=(0,...,0,q 1,...,q 1,q ). 1 1 If {wt(i),wt( j)} = { 1, + 1}, then supp(ql i) supp( j) = 0. Hence, there is at least one entry in (q l i+ j) which is not equal to q 1. If {wt(i),wt( j)}={ }, then supp(ql i) supp( j) =0. Hence, q l i and j ust have the following f i=(q 1,...,q 1,0,...,0), j=(0,...,0,q 1,...,q 1). 1

6 6 Therefe, the conclusion follows. 3) The proof is siilar to that of ) and is oitted here. As a consequence, we have the following proposition. Proposition 3. Let and n=q 1. 1) Suppose is odd. Then ) Suppose is even and q>. Then 3) Suppose is even and q=. Then {(cl(i),cl( j)) j C i,1 i, j l} 0 if 1 l q (+1)/ q, = h if l = q (+1)/ q+h, 1 h q, (q 1) if q (+1)/ 1 l q (+1)/. {(cl(i),cl( j)) j C i,1 i, j l} 0 if 1 l q /, 1 if q / 1 l q / 3, = if l = q /, 4 if q / 1 l q /. {(cl(i),cl( j)) j C i,1 i, j l} 0 if 1 l /, 1 if / 1 l (/)+1 4, = 3 if (/)+1 3 l (/)+1, 5 if (/)+1 1 l (/)+1. Cobining Lea 1 and Proposition 3, we have the following thee. Thee 4. Let and n=q 1. Given an integer δ (n+)/, let δ q and δ 0 be the unique integers such that δ 1=δ q q+δ 0, where 0 δ 0 < q. Then the diension k of the code C (q,,δ) is given below. 1) When is odd, { q (δ q (q 1)+δ 0 ) if δ q (+1)/ q, k= q (q ()/ 1)(q 1) if q (+1)/ q+1 δ q (+1)/ + 1. ) When is even and q>, q (δ q (q 1)+δ 0 ) if δ q / 1, q (δ q (q 1)+δ 0 1 ) if q/ δ q / + 1, k= q (δ q (q 1)+δ 0 1) if q / + δ q /, q (δ q (q 1)+δ 0 3 ) if δ=q/ 1, q (δ q (q 1)+δ 0 5 ) if q/ δ q / ) When is even and q=, (δ q + δ 0 ) if δ / 1 and 4, (δ q + δ 0 1 ) if / δ / + 1 and 4, k= (δ q + δ 0 1) if / + δ (/)+1 3 and 4, (δ q + δ 0 ) if (/)+1 δ (/)+1 1 and 6, (δ q + δ 0 3) if (/)+1 δ (/) and 6.

7 7 In addition, the iniu distance d of the code satisfies that d δ. Proof: Let g (q,,δ) (x) be the generat polynoial of C (q,,δ). F the diension of the code, we only prove ) since the proofs of 1) and 3) are siilar. By ) of Lea 1, the degree of g (q,,δ) (x) equals where 1+(δ q (q 1)+δ 0 ) ε {(cl(i),cl( j)) j C i,1 i, j δ 1}, ε= { 0 if δ q / + 1, 1 if δ q / +. Together with ) of Proposition 3, we have 1+(δ q (q 1)+δ 0 ) if δ q / 1, 1+(δ q (q 1)+δ 0 1 ) if q/ δ q / + 1, deg(g (q,,δ) (x))= 1+(δ q (q 1)+δ 0 1) if q / + δ q /, 1+(δ q (q 1)+δ 0 3 ) if δ=q/ 1, 1+(δ q (q 1)+δ 0 5 ) if q/ δ q / + 1. Therefe, the conclusion on the diension in ) follows. Meover, by the BCH bound, C (q,,δ) has iniu distance d δ. Reark 1. F the code C (q,,δ), if δ i=0 ( ) n (q 1) i > q n k, (1) i then d δ by the sphere packing bound. Therefe, the knowledge on the diension of the code C (q,,δ) ay provide e precise infation on the iniu distance in soe cases. As an illustration, we use Thee 4 and the inequality (1) to get soe binary codes C (,,δ) with d = δ, which are listed in Table I. TABLE I SOME BINARY CODE C (,,δ) WITH d = δ δ {5,6,7} {3} {8,9,10,11,1,13} {3,5} {14,15,17,17,18,19} {3,5,7} {0} {3,5,7,9} B. The diension of C (q,,δ) when δ=q λ and λ In [18], the diension of the narrow-sense priitive BCH code C (q,,δ) with δ=q λ is considered. The auth derived two closed fulas concerning the diension of such code. In this subsection, we use the idea in [18] to give an estiate of the diension of the reversible BCH code C (q,,δ) with δ=q λ, where λ. Let s and r be two positive integers. Given a sequence of length s and a fixed integer a with 0 a q 1, we say that the sequence contains a straight run of length r with respect to a, if it has r consecutive entries fed by a. If we view the sequence as a circle where the first and last entry are glued together, we say that the sequence contains a circular run of length r with respect to a, if this circle has r consecutive entries fed by a. When the specific choice of the integer a does not atter, we siply say that the sequence has a straight circular run of length r. Clearly, a straight run is also a circular run and the

8 8 converse is not necessarily true. We use l r (s) to denote the nuber of sequences of length s, which contains a straight run of length r. Particularly, we define l r (0)=0. The following is a recursive fula of l r (s) which was presented in [18]. Proposition 5. [18, p. 155] Let s and r be two nonnegative integers. Then 0 if 0 s<r, l r (s)= 1 if s=r, ql r (s 1)+(q 1)(q s r 1 l r (s r 1)) if s>r. Throughout the rest of this section, we always assue that δ = q λ and λ. Recall that the narrow-sense BCH code C (q,,δ) has generat polynoial g + (q,,δ)(x). Set r = λ. Note that δ 1 cresponds to following sequence δ 1=(0,...,0,q 1,q 1,...,q 1). }{{} λ 1 r The key observation in [18] is that f 1 i n 1, α i is a root of g + (q,,δ)(x) if and only if the sequence of i has a circular run of length at least r with respect to 0. Siilarly, note that n δ+1 cresponds to the following sequence n δ+1=(q 1,...,q 1, 0,0,...,0). }{{} λ 1 r Therefe, f 1 i n 1, α i is a root of g (q,,δ)(x) if and only if the sequence of i has a circular run of length at least r with respect to q 1. The following proposition presents the degree of g + (q,,δ)(x) and g (q,,δ) (x). Proposition 6. [18, p. 155] Set r= λ. Then deg(g + (q,,δ) (x))=deg(g (q,,δ) (x))=l r() 1+(q 1) r (r u 1)(q r u l r ( r u )). u=0 We have the following estiation on the diension of C (q,,δ). Thee 7. Set r= λ. The diension k of C (q,,δ) satisfies and k q l r ()+l r ( r) (q 1) r (r u 1)(q r u l r ( r u )), u=0 k q l r ()+l r ( r) (q 1) r (r u 1)(q r u l r ( r u )). u=0 Proof: Since λ, we have r. Define a set N ={1 i n 1 g+ (q,,δ) (αi )=g (q,,δ) (αi )=0}. Since g (q,,δ) (x)=lc(g (q,,δ) (x),x 1,g+ (q,,δ)(x)), we have deg(g (q,,δ) (x))=deg(g + (q,,δ) (x))+deg(g (q,,δ) (x))+1 N. Since deg(g + (q,,δ) (x)) and deg(g (q,,δ) (x)) are known by Proposition 6, it suffices to estiate the size of N. N contains the nuber 1 i n 1, such that i contains two runs of length r with respect to 0 and q 1, where at ost one of the is a circular run. Let N be the set of integers 1 i n such that the first r entries of i is a straight run of length r with respect to 0 and the last r entries contain a

9 9 straight run of length r with respect to q 1. Clearly, we have N =l r ( r). Note that each eleent of N is a proper cyclic shift of an eleent of N. Meover, f each i N, we have which iplies {q j i od n 0 j }, () N N N. Thus, the conclusion follows fro a direct coputation. III. THE MINIMUM DISTANCE OF THE REVERSIBLE BCH CODES C (q,,δ) While it is difficult to deterine the diension of reversible BCH codes in general, it is e difficult to find out the iniu distance of reversible BCH codes [8]. F the code C (q,,δ), the BCH bound d δ is usually very tight. But it would be better if we can deterine the iniu distance exactly. In this section, we deterine the iniu distance d of the code C (q,,δ) in soe special cases. Given a codewd c=(c 0,c 1,...,c n 1 ) C (q,,δ), we say c is reversible if (c n 1,c n,...,c 0 ) C (q,,δ). Naely, c C (q,,δ) is reversible if and only if c C (q,,δ). The following thee says that the reversible codewd in C (q,,δ) provides soe infation on the iniu distance on C (q,,δ). Thee 8. Let c(x) C (q,,δ) be a reversible codewd of weight w. If c(1) 0, then C (q,,δ) contains a codewd (x 1)c(x) whose weight is at ost w. Therefe the iniu distance d of C (q,,δ) satisfies d w. In particular, if the weight of c(x) is δ, then the iniu distance d = δ. Proof: Since c(x) is reversible and c(1) 0, we have (x 1)c(x) C (q,,δ). The weight of (x 1)c(x) is at ost w, which iplies d w. In particular, if w=δ, together with the BCH bound, we have d= δ. Let c(x)= i=0 n 1 c ix i be a codewd of a cyclic code C with length n=q 1. We can use the eleents of GF(q ) to index the coefficients of c(x). Siilarly, let C be the extended cyclic code of C and let c(x) = n i=0 c ix i be a codewd of C with length q. We can use the eleents of GF(q ) to index the coefficients of c(x). The suppt of c(x) (resp. c(x)) is defined to be the set of eleents in GF(q ) (resp. GF(q )), which crespond to the nonzero coefficients of c(x) (resp. c(x)). Given a prie power q and an integer 0 s q 1, s has a unique q-ary expansion s= i=0 s iq i. The q-weight of s is defined to be wt q (s)= i=0 s i. Suppose H is a subset of GF(q), then we use H ( 1) to denote the subset {h 1 h H}. The following are two classes of reversible BCH codes whose iniu distances are known. Collary 9. F the reversible BCH code C (q,,δ), we have d = δ if δ n. Proof: It suffices to find a codewd c(x) satisfying the condition in Thee 8. If δ n, by the proof of [4, Thee], C (q,,δ) contains a reversible codewd c(x) with weight δ, where c(x)= δ 1 i=0 a ix ni δ and c(1) 0. The desired conclusion then follows fro Thee 8. Collary 10. Let δ= r 1 and V be an -diensional vect space over GF(). Suppose 1 r, then we can choose four r-diensional subspaces of V, say H i, 1 i 4 of V, such that H 1 H ={0} and H 3 H 4 ={0}. If ((H 1 H )\{0}) ( 1) =(H 3 H 4 )\{0}, then the reversible BCH code C (,,δ) has iniu distance d = δ. Proof: Define C + (,,δ) (resp. C (,,δ)) to be the BCH code with length n and generat polynoial g + (,,δ) (x) (resp. g (,,δ) (x)). Let α be a priitive eleent of GF( ). We can assue the zeros of C + (,,δ) (resp. C (,,δ) ) include the eleents {αi 1 i δ 1} (resp. {α i 1 i δ 1}). The BCH code C + (,,δ) (resp. C (,,δ) ) contains a punctured Reed-Muller code RM+ ( r,) (resp. RM ( r,) ) as a subcode, in which RM + ( r,) has zeros {α i 0<i< 1,wt (i)<r}

10 10 and RM ( r,) has zeros {α i 0<i< 1,wt (i)<r}. Let c=(c 0,c 1,...,c n 1 ) be a codewd of RM + ( r,). Since RM + ( r,) is a cyclic code, its codinates can be indexed in the following way c=(c 0 1,c 1 α,...,c n 1 α n 1 ), (3) where n 1 j=0 c jα i j = 0 f each 1 i δ 1. Siilarly, suppose c =(c 0,c 1,...,c n 1 ) is a codewd of RM ( r,). Then, its codinates can be indexed in the following way c =(c 0 1, c 1 α 1,..., c n 1 α (n 1) ), (4) where n 1 j=0 c j α i j = 0 f each 1 i δ 1. By [, Collary 5.3.3], RM + ( r,) contains two iniu weight codewds c 1 (x) and c (x), such that the suppt of c 1 (x) and c (x) are H 1 \{0} and H \{0} respectively. Siilarly, RM ( r,) contains two iniu weight codewds c 3 (x) and c 4 (x), such that the suppt of c 3 (x) and c 4 (x) are H 3 \{0} and H 4 \{0} respectively. Meover, the codinates of c 1 (x) and c (x) are arranged in the way of (3) and the codinates of c 3 (x) and c 4 (x) are arranged in the way of (4). Therefe, c 1 (x)+c (x) RM + ( r,) C + (,,δ) and c 3 (x)+c 4 (x) RM ( r,) C (,,δ). Since ((H 1 H )\{0}) ( 1) =(H 3 H 4 )\{0}, by the arrangeent of the codinates of c i (x), 1 i 4, the two codewds c 1 (x)+c (x) and c 3 (x)+c 4 (x) coincide. Thus, we have c 1 (x)+c (x) C (,,δ). Since c 1 (1)+c (1)=0, we have a codewd c 1 (x)+c (x) C (,,δ) with weight δ. Exaple 1. Let q =, = 5 and δ = 3. We consider the iniu distance of C (,5,3). Let α be a priitive eleent of GF( 5 ) and the inial polynoial of α over GF() is x 5 + x + 1. Then we have the following four -diensional subspaces of GF( 5 ): H 1 ={0,α,α,α 19 }, H ={0,α 8,α 1,α 18 }, H 3 ={0,α 1,α 13,α 30 }, H 4 ={0,α 19,α 3,α 9 }. Thus, we have c 1 (x) and c (x) as codewds of C + (,5,3), whose suppts are H 1\{0} and H \{0}. We have c 3 (x) and c 4 (x) as codewds of C (,5,3), whose suppts are H 3\{0} and H 4 \{0}. Clearly, ((H 1 H )\{0}) ( 1) =(H 3 H 4 )\{0}. Therefe, c 1 (x)+c (x) coincides with c 3 (x)+c 4 (x), whose weight is six. Consequently, c 1 (x)+c (x) C (,5,3) and the iniu distance of C (,5,3) equals 6. Based on our nuerical experient, we have the following conjecture, which can be regarded as an analogy of [17, Chapter 9, Thee 5]. Conjecture 1. Let δ=q λ 1, where 1 λ /. Then the code C (q,,δ) has iniu distance d= δ.

11 11 IV. THE PARAMETERS OF THE REVERSIBLE CODE C (q,,δ) FOR SMALL δ In this section, we deterine the paraeters of the code C (q,,δ) f a few sall values of δ. With the help of Thee 4 and Collary 9, we can achieve this in soe cases. Recall that the Melas code over GF(q) is a cyclic code with length n=q 1 and generat polynoial 1 (x) 1 (x) and was first studied by Melas f the case q= [1]. The weight distribution of the Melas code has been obtained f q=,3 [14], []. F δ=, the code C (q,,δ) is the even-like subcode of the Melas code. The following thee is a direct consequence of Thee 4 and Collary 9. Thee 11. Suppose q is odd and, then C (q,,) has paraeters [q 1,q,4]. When δ=3, we have the following result. Thee 1. 1) When q= and 4, C (q,,3) has paraeters [ 1,,6]. ) When q 1 (od 3) and 4, C (q,,3) has paraeters [q 1,q 4,6]. Proof: 1) The diension follows fro Thee 4. Applying the BCH and the sphere packing bound, we can see that the iniu distance is 6. ) The diension follows fro Thee 4. Since q 1 (od 3), we have 3 n. Therefe, by Collary 9, the iniu distance is 6. Thee 13. Suppose 3, then C (3,,4) has paraeters [3 1,3 4,d], where d = 8 if is even and d 8 if is odd. Proof: It follows fro Thee 4 that the diension of this code is equal to q 4. By the BCH bound, the iniu distance of C (3,,4) is at least 8. When is even, 4 divides n. Hence, the iniu distance of C (3,,4) is equal to 8 accding to Collary 9. We have the following conjecture concerning the case q=3. Conjecture. When δ=4, 3 is odd, and q=3, C (q,,δ) has iniu distance d = 8. Exaple. Let =3 and q=3. Let α be a generat of GF(3 3 ) with α 3 α+1 = 0. Then C (q,,δ) has paraeters [6, 13, 8] and generat polynoial x 13 + x 1 + x 11 + x 10 + x 8 + x 5 + x 3 + x + x+. This is the best ternary cyclic code and has the sae paraeters as the best linear code with the sae length and diension. V. CONCLUDING REMARKS In this paper, we deterined the diension of the reversible BCH code C (q,,δ) when δ is relatively sall, and developed lower and upper bounds on its diension f δ = q λ, where λ. We deterined the iniu distance of C (q,,δ) in two special cases. Meover, we settled the paraeters of the code C (q,,δ) f soe very sall δ. However, the paraeters of the reversible BCH codes C (q,,δ) are still open in ost cases. It would be nice if further progress on the deterination of the paraeters of the codes C (q,,δ) can be ade. Accding to the tables of best cyclic codes docuented in [10], the binary reversible BCH code C (q,,δ) is the best possible cyclic code in alost all cases (see Table II). Hence, it is wthwhile to study the code C (q,,δ).

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