Constacyclic symbol-pair codes: lower bounds and optimal constructions
aa r X i v : . [ c s . I T ] M a y Constacyclic symbol-pair codes: lower bounds and optimalconstructions ∗ Bocong Chen , Liren Lin , Hongwei Liu School of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510641, China School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei, 430079, China
Abstract
Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pairerrors in symbol-pair read channels. The higher the minimum pair distance, the more pair errorsthe code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot beimproved for given length and code size. The contribution of this paper is twofold. First we presentthree lower bounds for the minimum pair distance of constacyclic codes, the first two of whichgeneralize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015).The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes.Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight throughrepeated-root cyclic codes.
Keywords:
Symbol-pair read channel, symbol-pair code, MDS symbol-pair code, repeated-rootcyclic code.
Let Σ be a set of size q , which we refer to as an alphabet and whose elements are called symbols . A q -arycode C of length n over Σ is a nonempty subset of Σ n . For any vector a = ( a , a , · · · , a n − ) ∈ Σ n , the symbol-pair read vector of a is defined to be π ( a ) = (cid:2) ( a , a ) , ( a , a ) , · · · , ( a n − , a n − ) , ( a n − , a ) (cid:3) . Two pairs ( c, d ) and ( e, f ) are distinct if c = e or d = f , or both. The pair distance between a and b ,denoted by d p ( a , b ), is defined as d p (cid:0) a , b (cid:1) = d H (cid:0) π ( a ) , π ( b ) (cid:1) , where d H denotes the usual Hamming distance. It turns out that the set Σ n equipped with the pairdistance d p is indeed a metric space (see [3]). In a similar way to Hamming-metric codes, the minimumpair distance of a code C is defined to be d p (cid:0) C (cid:1) = min (cid:8) d p (cid:0) a , b (cid:1) (cid:12)(cid:12) a , b ∈ C , a = b (cid:9) . For any code C of length n with 0 < d H ( C ) < n , a simple but important connection between d H ( C ) and d p ( C ) is given in [3]: d H ( C ) + 1 ≤ d p ( C ) ≤ d H ( C ). A code of length n over Σ is called an ( n, M, d p ) q -symbol-pair code if its size is M and minimum pair distance is d p .Symbol-pair codes introduced by Cassuto and Blaum [2, 3] are designed to protect against pair errorsin symbol-pair read channels, where the outputs are overlapping pairs of symbols. The seminal works[2, 3, 4] have established relationships between the minimum Hamming distance of an error-correcting ∗ E-Mail addresses: bocong [email protected] (B. Chen), L R [email protected] (L. Lin), [email protected] (H. Liu). d p then it can correct up to ⌊ ( d p − / ⌋ symbol-pair errors. For this reason, it is desirable toconstruct symbol-pair codes having a large minimum pair distance.For a fixed code length n , it would certainly be nice if both the code size M (which is a measure ofthe efficiency of the code) and the minimum pair distance d p could be as large as possible. However, asin the Hamming-metric case, these two parameters are restricted each other for any fixed length. TheSingleton-type Bound for symbol-pair codes relates the parameters n , M and d p (see [6, Theorem 2.1]):If C is an ( n, M, d p ) q -symbol-pair code with q ≥ ≤ d p ≤ n , then M ≤ q n − d p +2 . (1.1)A symbol-pair code for which equality holds in (1.1) is said to be maximum distance separable (MDS).In this case, the code size M is fully determined by n, d p and q . Following [6], we use ( n, d p ) q to denotean MDS symbol-pair code of length n over Σ with minimum pair distance d p and size M = q n − d p +2 .MDS symbol-pair codes are optimal in the sense that no code of length n with M codewords has a largerminimum pair distance than an MDS symbol-pair code with parameters n and M . Constructing MDSsymbol-pair codes is thus of significance in theory and practice.Cassuto and Blaum [3] studied how the class of cyclic codes can be exploited as a framework forsymbol-pair codes. Combining the discrete Fourier transform (DFT) with the BCH Bound, [3, Theorem10] showed that if the generator polynomial of a simple-root [ n, k, d H ] cyclic code has at least d H roots(in some extension field over F q ), then the minimum pair distance of the code is at least d H + 2. Usingthe Hartmann-Tzeng Bound, this lower bound was improved to d H + 3 when the code length n is a primenumber and a constraint condition on n, k and d H is assumed (see [3, Theorem 11]). In a follow-uppaper [12], Kai et al. showed that [3, Theorem 10] can be generalized to simple-root constacyclic codes:If the generator polynomial of a simple-root [ n, k, d H ] constacyclic code has at least d H roots, then theminimum pair distance of the code is at least d H + 2 (see [12, Lemma 4.1]). Recently, Yaakobi et al. [13, Theorem 4] obtained an elegant result on the minimum pair distance of binary cyclic codes: If C isa binary cyclic code of dimension greater than one, then d p ( C ) ≥ d H ( C ) + ⌈ d H ( C )2 ⌉ . After establishing the Singleton Bound (1.1) for symbol-pair codes, Chee et al. [6, 7] employed variousmethods to construct MDS symbol-pair codes, including the use of classical MDS codes, interleavingmethod of Cassuto and Blaum [3], and eulerian graphs of certain girth, etc. It is worth noting that incontrast with all known classical MDS codes, of which the lengths are so small with respect to the alphabetsize, MDS symbol-pair codes can have relatively large code length (see [6]). In the light of the SingletonBound (1.1) and [12, Lemma 4.1], Kai et al. [12] used almost MDS constacyclic codes to construct MDSsymbol-pair codes; several classes of almost MDS constacyclic codes with minimum Hamming distancethree or four are constructed, and, consequently, MDS symbol-pair codes with minimum pair distancefive or six are obtained.The aforementioned works lead us to the study of lower bounds for the minimum pair distance ofconstacyclic codes and constructions of MDS symbol-pair codes. The contribution of this paper is twofold.First we present three lower bounds for the minimum pair distance of constacyclic codes, the first twoof which generalize the previously known results [3, Theorem 10], [3, Theorem 11] and [12, Lemma4.1]. The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes.Second we construct new MDS symbol-pair codes with minimum pair distance seven and eight by usingrepeated-root cyclic codes. More precisely, we summarize our results as follows.Thereafter, F q denotes a finite field of size q , where q is a power of a prime number p . Let n > n and p are not necessarily co-prime). Theorem 1.1.
Let C be an [ n, k, d H ] constacyclic code over F q with ≤ d H < n . Then we have thefollowing. (1) d p ( C ) ≥ d H +2 if and only if C is not an MDS code, i.e., k < n − d H +1 . Equivalently, d p ( C ) = d H +1 if and only if C is an MDS code, i.e., k = n − d H + 1 . (2) If k > and n − d H ≥ k − , then d p ( C ) ≥ d H + 3 . heorem 1.2. Let D be a nonzero [ ℓp e , k, d H ] repeated-root cyclic code over F q with generator polynomial g ( x ) , where ℓ > is a positive integer co-prime to p and e is a positive integer. If d H ( D ) is a primenumber and if one of the following two conditions is satisfied (1) ℓ < d H ( D ) < ℓp e − k ; (2) x ℓ − is a divisor of g ( x ) and < d H ( D ) < ℓp e − k ,then d p ( D ) ≥ d H ( D ) + 3 . At this point we make several remarks. The first part of Theorem 1.1 extends [3, Theorem 10] and [12,Lemma 4.1] in two directions: First we improve the results by giving a necessary and sufficient condition.Second we do not require that gcd( n, q ) = 1.We make a comparison between [3, Theorem 11] and the second part of Theorem 1.1. [3, Theorem 11]says that if a q -ary [ n, k, d H ] simple-root cyclic code with prime length n satisfies n − d H ≥ k −
2, thenthe minimum pair distance of the code is at least d H + 3. The second part of Theorem 1.1 removes theprime-length constraint and the simple-root requirement; if n − d H is odd, the conditions n − d H ≥ k − n − d H ≥ k − k ≤ ( n − d H ) / k ≤ ( n − d H ) / Theorem 1.3.
The following hold. (1)
Let p ≥ be an odd prime number. Then there exists an MDS (3 p, p -symbol-pair code. (2) Let p be an odd prime number such that is a divisor of p − . Then there exists an MDS (3 p, p -symbol-pair code. (3) Let p ≥ be an odd prime number. Then there exists an MDS (3 p, p -symbol-pair code. (4) Let q ≥ be a prime power and let n ≥ q + 4 be a divisor of q − . Then there exists an MDS (cid:0) n, (cid:1) q -symbol-pair code. Note that [12, Theorem 4.3] asserts that there exists an MDS (cid:0) n, (cid:1) q -symbol-pair code if n > q + 1is a divisor of q −
1. The fourth part of Theorem 1.3 shows that the minimum pair distance 5 can beincreased to 6.This paper is organized as follows. Basic notations and results about constacyclic codes and repeated-root cyclic codes are provided in Section 2. The proofs of Theorems 1.1 and 1.2, together with somecorollaries and examples, are presented in Section 3. The proof of Theorem 1.3 is given in Section 4.
In this section, basic notations and results about constacyclic codes and repeated-root cyclic codes areprovided. The result [5, Theorem 1] plays an important role in the proof of Theorems 1.2 and 1.3, whichprovides an effective way to determine the minimum Hamming distance of repeated-root cyclic codes.A code C of length n over F q is a nonempty subset of F nq . If, in addition, C is a linear subspaceover F q of F nq , then C is called a linear code . A linear code C of length n , dimension k and minimumHamming distance d H over F q is often called a q -ary [ n, k, d H ] code. Given a nonzero element λ of F q ,the λ -constacyclic shift τ λ on F nq is the shift τ λ (cid:0) ( x , x , . . . , x n − ) (cid:1) = (cid:0) λx n − , x , x , . . . , x n − (cid:1) . A linear code C is said to be λ -constacyclic if C is a τ λ -invariant subspace of F nq , i.e., τ λ ( C ) = C . Inparticular, it is just the usual cyclic code when λ = 1. In studying constacyclic codes of length n , itis convenient to label the coordinate positions as 0 , , · · · , n −
1. Since a constacyclic code of length n contains all n constacyclic shifts of any codeword, it is convenient to think of the coordinate positions3yclically where, once you reach n −
1, you begin again with coordinate 0. When we speak of consecutivecoordinates, we will always mean consecutive in that cyclical sense.Each codeword c = ( c , c , . . . , c n − ) ∈ C is customarily identified with its polynomial representation c ( x ) = c + c x + · · · + c n − x n − . Any code C is then in turn identified with the set of all polynomialrepresentations of its codewords. In this way, a linear code C is λ -constacyclic if and only if it is an idealof the quotient ring F q [ x ] / h x n − λ i (e.g., see [8]). It follows that a unique monic divisor g ( x ) ∈ F q [ x ] of x n − λ can be found such that C = h g ( x ) i = (cid:8) f ( x ) g ( x ) (mod x n − λ ) (cid:12)(cid:12) f ( x ) ∈ F q [ x ] (cid:9) . The polynomial g ( x ) is called the generator polynomial of C , in which case C has dimension k precisely when the degreeof g ( x ) is n − k .Generally, constacyclic codes over finite fields can be divided into two classes: simple-root constacycliccodes, if the code lengths are co-prime to the characteristic of the field; otherwise, we have the so-calledrepeated-root constacyclic codes. Most of studies on constacyclic codes in the literature are focused onthe simple-root case, which essentially guarantees that every root of x n − λ has multiplicity one. Simple-root constacyclic codes are thus can be characterized by their defining sets (e.g., see [9] or [11]). TheBCH Bound and the Hartmann-Tzeng Bound for simple-root cyclic codes (e.g., see [10]) are based onconsecutive sequences of roots of the generator polynomial.In contrast to the simple-root case, repeated-root constacyclic codes are no longer characterized bysets of zeros. Castagnoli et al. [5, Theorem 1] determined the minimum Hamming distance of repeated-root cyclic codes by using polynomial algebra; it is showed that the minimum Hamming distance ofa repeated-root cyclic code D can be expressed in terms of d H ( ¯ D t ), where ¯ D t are simple-root cycliccodes fully determined by D . To include [5, Theorem 1], we first introduce the following notation. Let D = h g ( x ) i be a repeated-root cyclic code of length ℓp e over F q , where ℓ > ℓ, p ) = 1 and e is a positive integer. Suppose g ( x ) = s Y i =1 m i ( x ) e i is the factorization of g ( x ) into distinct monic irreducible polynomials m i ( x ) ∈ F q [ x ] of multiplicity e i .Fix a value t , 0 ≤ t ≤ p e −
1; ¯ D t is defined to be a (simple-root) cyclic code of length ℓ over F q withgenerator polynomial g t ( x ) as the product of those irreducible factors m i ( x ) of g ( x ) that occur withmultiplicity e i > t . If this product turns out to be x ℓ −
1, then ¯ D t contains only the all-zero codewordand we set d H ( ¯ D t ) = ∞ . If all e i (1 ≤ i ≤ s ) satisfy e i ≤ t , then, by way of convention, g t ( x ) = 1 and d H ( ¯ D t ) = 1. The next result is an immediate consequence of [5, Lemma 1] and [5, Theorem 1]. Lemma 2.1.
Let D = h g ( x ) i be a repeated-root cyclic code of length ℓp e over F q , where ℓ > is a positiveinteger such that gcd( ℓ, p ) = 1 and e is a positive integer. Then d H ( D ) = min (cid:8) P t · d H ( ¯ D t ) (cid:12)(cid:12) ≤ t ≤ p e − (cid:9) where P t = Y i (cid:0) t i + 1 (cid:1) (2.1) with t i ’s being the coefficients of the radix- p expansion of t . The proof of Theorem 1.1 is given below.
Proof of Theorem 1.1.
To prove (1), we first observe that d p ( C ) ≥ d H +1 since the minimum Hammingdistance of C satisfies 2 ≤ d H < n . We will show that d p ( C ) = d H + 1 if and only if k = n − d H + 1. Tothis end, we claim that d p ( C ) = d H + 1 precisely when C has a codeword with Hamming weight d H inthe form (cid:0) a i , a i , · · · , a i d , , · · · , (cid:1) , a i j are nonzero elements of F q for 1 ≤ j ≤ d (here, d H is denoted by d for short). Indeed, it isclear that d p ( C ) = d H + 1 if and only if there exists a codeword c ∈ C such that w H ( c ) = d H and the d H nonzero terms appear with consecutive coordinates; applying the λ -constacyclic shift a certain numberof times on c if necessary, c is then converted to the form ( a i , · · · , a i d , , · · · , C is an MDS code, namely k = n − d H + 1, then d p ( C ) = d H + 1. For the converse, let H =( h , · · · , h n ) be a parity-check matrix for C , where h i (1 ≤ i ≤ n ) are the columns of H . Suppose d p ( C ) = d + 1, then there exists a codeword c = ( a i , · · · , a i d , , · · · , ∈ C , as claimed in the preceding paragraph.Hence, a i h + · · · + a i d h d = 0, which implies that the d th column h d lies in the ( d − F n − kq spanned by h , h , · · · , h d − , say V = h h , · · · , h d − i . Using the λ -constacyclic shift on c , it follows that (0 , a i , · · · , a i d , , · · · ,
0) is also a codeword of C . Therefore, a i h + · · · + a i d h d +1 = 0.This leads to h d +1 ∈ V . We can continue in this fashion and eventually obtain that the dimension of thevector space generated by the columns of H is exactly equal to d H −
1. However, H is a full row-rankmatrix of size ( n − k ) × n , which forces n − k = d H −
1. This completes the proof of (1).The proof of Theorem 1.1(2) needs the following corollary. Using essentially identical arguments tothe proof Theorem 1.1(1), we have the following result.
Corollary 3.1.
Let C be an [ n, k, d H ] constacyclic code over F q with ≤ d H < n . If C contains acodeword, of which the Hamming weight is d H + 1 , such that the d H + 1 nonzero terms appear withconsecutive coordinates, then n − d H ≤ k . Now we continue to give the proof of Theorem 1.1(2). Since the parameters of C satisfy n − d H ≥ k − d p ( C ) ≥ d H + 2. In order to prove d p ( C ) ≥ d H + 3, it suffices toshow that there are no codewords of C with Hamming weight d H + 1 such that the d H + 1 nonzero termsappear with consecutive coordinates, and that there are no codewords of C with Hamming weight d H inthe form ( a , r , b , s ), where a , b are row vectors with all the entries of a , b being nonzero, r and s are all-zero row vectors of lengths r and s respectively.From k > n − d H ≥ k −
1, we see that n − d H > k . Using Corollary 3.1, we are left toshow that there are no codewords of C with Hamming weight d H in the form ( a , r , b , s ). Supposeotherwise that c = ( a , r , b , s ) ∈ C with w H ( c ) = d H . We will derive a contradiction. Let g ( x ) be thegenerator polynomial of C . Then there exists a unique polynomial u ( x ) with deg u ( x ) ≤ k − u ( x ) g ( x ) = c ( x ) = ( a , r , b , s ). If s ≥ k , then the degree of c ( x ) is at most n − k −
1. This is impossiblebecause the degree of g ( x ) is n − k . We thus conclude that s ≤ k −
1. Similar reasoning yields r ≤ k − n − d H = r + s ≤ k −
2, which contradicts the hypotheses of the theorem. We are done.We illustrate Theorem 1.1 in the following example.
Example 3.2.
Take q = 5 and n = 24 in Theorem 1.1. Let C be a cyclic code of length 24 over F withdefining set T = Z \ { , , } . Magma [1] computations show that C has parameters [24 , , −
19 = 5 = 2 × −
1, it follows from Theorem 1.1(2) that d p ( C ) ≥
19 + 3 = 22. In fact, C has minimumpair distance 23, which gives that C is an MDS (24 , -symbol-pair code.We now turn to the proof of Theorem 1.2. Proof of Theorem 1.2.
It follows from Theorem 1.1 that d p ( D ) ≥ d H ( D ) + 2. By d H ( D ) < ℓp e − k again, Corollary 3.1 ensures that there are no codewords of D with Hamming weight d H + 1 such thatthe d H + 1 nonzero terms appear with consecutive coordinates. Therefore, it remains to show that thereare no codewords of D with Hamming weight d H in the form (cid:0) a , r , b , s (cid:1) (3.1)where a , b are row vectors with all the entries of a , b being nonzero, r and s are all-zero row vectorsof lengths r and s respectively.To this end, we first analyze the nonzero codewords of D by using [5, Lemma 2]. Let c ( x ) ∈ D bean arbitrary nonzero codeword of degree at most ℓp e −
1. Write c ( x ) as c ( x ) = ( x ℓ − t v ( x ), where0 ≤ t ≤ p e − x ℓ − v ( x ), and write v ( x ) in the form v ( x ) = v ( x ℓ ) + xv ( x ℓ ) + · · · + x ℓ − v ℓ − ( x ℓ ) . (3.2)55, Lemma 2] says that c ¯ t ( x ) = ( x ℓ − ¯ t ¯ v ( x ) p e (mod x ℓp e − , where ¯ v ( x ) ≡ v ( x ) (mod x ℓ −
1) and ¯ t = min { ¯ t ∈ T | ¯ t ≥ t } (The elements of T are nonnegativeintegers; for the definition of T , the reader may refer to [5]), is also a nonzero codeword of D satisfying w H ( c ¯ t ( x )) ≤ w H ( c ( x )).Now choosing c ( x ) to be any codeword of D with Hamming weight d H , [5, Lemma 2] and [5, Theorem1] together with their proofs tell us more: d H = w H (cid:0) c ( x ) (cid:1) = w H (cid:0) c ¯ t ( x ) (cid:1) = P ¯ t · N v , where N v is the number of nonzero v i ( x ℓ )’s in (3.2) and P ¯ t is a positive integer defined in (2.1). Thesefacts yield d H = P ¯ t or d H = N v , with our assumption that d H is a prime number. If (1) holds, we have N v = 1 since N v ≤ ℓ ; if (2) holds, it follows from t ≥ t ≥
1, and thus P ¯ t ≥ N v = 1.In conclusion, c ( x ) must be one of the following forms: c ( x ) = x i ( x ℓ − t v i ( x ℓ ) for some 0 ≤ i ≤ ℓ − . Expanding c ( x ) and using the fact that the degree of c ( x ) is at most ℓp e −
1, it follows from d H ( D ) ≥ c ( x ) cannot have the form (3.1). This completes the proof.We give two examples to illustrate Theorem 1.2. Example 3.3.
Take ℓ = 3, p = 5 and e = 1 in Theorem 1.2. Let D be a repeated-root cyclic code oflength 15 over F with generator polynomial ( x − x − D has parameters[15 , , D is at least 6. Now the Singleton Bound for symbol-pair codes (1.1) gives that D is anMDS (15 , -symbol-pair code.Example 3.3 suggests an infinite family of MDS symbol-pair codes with minimum pair distance six aswe show below. Corollary 3.4.
Let p ≥ be an odd prime number. Then there exists an MDS (3 p, p -symbol-pair code.Proof. Let D be a repeated-root cyclic code of length 3 p over F p with generator polynomial ( x − x − d H ( D ) = 3, and so D has parameters [3 p, p − , Example 3.5.
Take ℓ = 3, p = 7 and e = 1 in Theorem 1.2. Let D be a repeated-root cyclic code oflength 21 over F with generator polynomial ( x − ( x − ( x − D has parameters [21 , , D is at least 8. Magma [1] computations show that (6 , , , , , , , ), where and denoterespectively all-zero row vectors of length 6 and 9, is a codeword of D . Therefore, the true minimum pairdistance of D is 8. The proof of Theorem 1.3 is presented as follows.
Proof of Theorem 1.3. (1) . Let D be a cyclic code of length 3 p over F p with generator polynomial g ( x ) = ( x − ( x + x + 1). Using Lemma 2.1, we have that D is a cyclic code over F p with parameters[3 p, p − , d p ( D ) ≥ D with Hamming weight 5 such that the 5 nonzero terms appear with consecutive coordinates. We are leftto show that there are no codewords of D with Hamming weight 4 in the form (cid:0) a , u , b , v (cid:1) (4.1)6here a , b are row vectors with all the entries of a , b being nonzero, u and v are all-zero row vectors oflengths u and v respectively. Let c ( x ) be a minimum Hamming weight codeword of D of degree at most3 p −
1. Write c ( x ) as c ( x ) = ( x − t v ( x ) where 0 ≤ t ≤ p − x − v ( x ), andwrite v ( x ) in the form v ( x ) = v ( x ) + xv ( x ) + x v ( x ) . (4.2)Since x − g ( x ), we have t ≥
1. As pointed out in the proof ofTheorem 1.2, the following equalities hold:4 = w H (cid:0) c ( x ) (cid:1) = (1 + t ) · N v , (4.3)where N v is the number of nonzero v i ( x )’s in (4.2). There are two possible values for N v : If t = 3, then N v = 1; if t = 1, then N v = 2. The case N v = 1 clearly implies that c ( x ) cannot be in the form (4.1). Thuswe only need to consider the case N v = 2 and t = 1. Assume to the contrary that c ( x ) = ( x − v ( x ) is aminimum Hamming weight codeword of D in the form (4.1). Without loss of generality we may supposethat the first coordinate of c ( x ) is 1. There are two cases:Case 1: v ( x ) = v ( x ) + xv ( x ). The forms of ( x − v ( x ) and x ( x − v ( x ) can be illustratedby the following table:( x − v ( x ) 1 0 0 ✷ ✷ · · · ✷ x ( x − v ( x ) 0 ✷ ✷ ✷ · · · ✷ ✷ marks the possible nonzero terms. To ensure that c ( x ) is in the form (4.1), theHamming weight of ( x − v ( x ) must be equal to 2 and the coefficient of x in the expansion of x ( x − v ( x ) must be nonzero. Therefore, a positive integer r with 1 ≤ r ≤ p − a , a , a of F p can be found such that c ( x ) = 1 + a x + a x r + a x r +1 . With c (1) = c ( ω ) = c ( ω ) = 0, we have a = − a and a = −
1. On the other hand, the first and thesecond formal derivative of c ( x ) respectively gives c (1) ( x ) = a − rx r − − (3 r + 1) a x r and c (2) ( x ) = − r (3 r − x r − − r (3 r + 1) a x r − . Since ( x − is a divisor of c ( x ), it follows from c (1) (1) = c (2) (1) = 0 that a = − r = 0. This isa contradiction, for p ≥ ≤ r ≤ p − v ( x ) = v ( x ) + x v ( x ). As in the previous case, a positive integer r with 1 ≤ r ≤ p − a , a , a of F p can be found such that c ( x ) = 1 + a x r − + a x r + a x p − . With arguments similar to the previous case, we have 6 r = 0, a contradiction again. This completes theproof of the first part of Theorem 1.3.(2) . Let D be a repeated-root cyclic code of length 3 p over F p with generator polynomial ( x − ( x − ω ) ( x − ω ). Using Lemma 2.1, we have that D is a cyclic code over F p with parameters [3 p, p − , d p ( D ) ≥
6. Using techniques similar to those used in the proof of Theorem 1.1, wesee that there are no codewords of D with Hamming weight 5 (resp. 6) such that the 5 (resp. 6) nonzeroterms appear with consecutive coordinates.The proof will be completed in three steps.Step 1. There are no codewords of D with Hamming weight 4 in the form (cid:0) a , u , b , v (cid:1) where a , b are row vectors with all the entries of a , b being nonzero, u and v are all-zero row vectorsof lengths u and v respectively. It is trivial to see that this holds by the same arguments as in the proofof (1). 7tep 2. There are no codewords of D with Hamming weight 4 in the form (cid:0) a , u , b , v , c , w (cid:1) . (4.4)where a , b and c are row vectors with all the entries of a , b and c being nonzero, u , v and w areall-zero row vectors of lengths u , v and w respectively. Assume to the contrary that c ( x ) = ( x − v ( x ) isa minimum Hamming weight codeword of D in the form (4.4). Without loss of generality we may supposethat the first coordinate of c ( x ) is 1. At this point, we arrive at (4.3) again. There are two possible valuesfor N v : If t = 3, then N v = 1; if t = 1, then N v = 2. Clearly, c ( x ) cannot be in the form (4.4) if N v = 1.We are left to consider the case N v = 2 and t = 1. We now consider three cases separately.Case 1. c ( x ) = 1 + a x + a x r + a x s , where r, s are positive integers with 1 ≤ r = s ≤ p − a , a , a are nonzero elements of F p . With c (1) = c ( ω ) = 0, we have1 + a + a + a = 0 and 1 + a ω + a + a = 0 , which forces a = 0, a contradiction.Case 2. c ( x ) = 1 + a x + a x r +1 + a x s +1 , where r, s are positive integers with 1 ≤ r = s ≤ p − a , a , a are nonzero elements of F p . It follows from c (1) = c ( ω ) that1 + a + a + a = 0and 1 + a ω + a ω + a ω = 0 . This is impossible.Case 3. c ( x ) = 1 + a x + a x r + a x s +1 , where r, s are positive integers with 1 ≤ r = s ≤ p − a , a , a are nonzero elements of F p . With c (1) (1) = c (1) ( ω ) = 0, we have a + 3 ra + (3 s + 1) a = 0and a + 3 rω a + (3 s + 1) a = 0 , a contradiction.Step 3. There are no codewords of D with Hamming weight 5 in the form (cid:0) a , u , b , v (cid:1) where a , b are row vectors with all the entries of a , b being nonzero, u and v are all-zero row vectorsof lengths u and v respectively. It is easy to see that this case holds.This completes the proof of the second part of Theorem 1.3.(3) . This has been done in Corollary 3.4.(4) . By our assumption q ≡ n ), every q -cyclotomic coset modulo n has size one or two.Clearly, the congruence q ( q + 1) ≡ q + 1 (mod n ) implies that the q -cyclotomic coset containing q + 1,denoted by C q +1 , has exactly one element. Let C be a cyclic code of length n over F q with defining set T = C S C S C q +1 , where C = { } , C = { , q } and C q +1 = { q + 1 } . It is easy to see that C hasdimension k = n − d H ( C ) is 4 by using the Hartmann-Tzeng Bound (see [10,Theorem 4.5.6]). Indeed, applying the Hartmann-Tzeng Bound with A = { , } and B = { , q } (sincegcd( q, n ) = 1), we obtain d H ( C ) ≥ d H ( C ) ≤ n − ( n −
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