A Class of Nonbinary Codes and Their Weight Distribution

A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION XIANGYO NG ZENG, NIAN LI, AND LEI HU Abstra ct. In this pap er, for an even in teger n ≥ 4 and any p ositive integer k with gcd( n/ 2 , k ) = gcd( n/ 2 − k, 2 k ) = d b eing o dd, a class of p -ary cod es C k is defined and their wei ght distribution is completely determined, where p is an o dd prime. As an application, a class of non binary sequen ce families is constructed from these cod es, and the correlation distribution is also determined. 1. Introduction Nonlinear functions ha v e imp ortant applications in cod ing theory and cryptograph y [16, 7]. Linear co des constr u cted from functions with high nonlinearit y [1 7, 11, 2, 6] can be g o o d and hav e u seful applications in comm un ications [9, 1 9, 1 0 , 18] or cr y p tograph y [5, 4, 3, 21]. F or a cod e, its weigh t distribution is im p ortant to study its stru cture and to provi de information on the probabilit y of undetected err or when the co d e is u s ed for error detection. Throughout this pap er, let F q b e the fin ite field with q = p n elemen ts f or a prime p and a p ositiv e in teger n , and let F ∗ q b e the multiplica tiv e group of F q . F or an ev en inte ger n ≥ 4, let C k denote the [ p n − 1 , 5 n/ 2] cyclic co de giv en by C k = n c ( γ , δ, ǫ ) =  Π γ ,δ ( x ) + T r n 1 ( ǫx )  x ∈ F ∗ p n | γ ∈ F p n/ 2 , δ , ǫ ∈ F p n o constructed fr om the function Π γ ,δ ( x ) = T r n/ 2 1 ( γ x p n/ 2 +1 ) + T r n 1 ( δ x p k +1 ) , (1.1) where 1 ≤ k < n w ith k 6 = n / 2, and for a p ositive integ er l , T r l 1 ( · ) is the trace fun ction from F p l to F p . Sev eral classes of binary co des C k ha ve b een extensiv ely studied for some v alues of the parameter k . The binary co de C n/ 2 ± 1 is exactly the Kasami co d e in Theorem 14 of [9]. B y c h o osing cyc licly inequiv alen t codewords from the Kasami co de, the large set of bin ary Kasami sequences w as ob- tained [19]. The minimum distance b ound of C 1 w as established b y ev aluating the exp on ential sums P x ∈ F 2 n ( − 1) Π γ ,δ ( x )+ T r n 1 ( ǫx ) in [15] and [12], and the w eight distribu tion and th en the minim um distance w ere completely determined in [20]. F or ev en n/ 2, the binary co d e C 1 has the same weig h t d istribution as th e K asami co de [9, 20]. F urther m ore, for any k w ith gc d ( k , n ) = 2 if n/ 2 is o dd or gcd( k , n ) = 1 if n/ 2 is eve n, the weigh t distribu tion of binary cod es C k w as also determined, and these co des were used to construct families of generalized Kasami sequences, whic h ha ve the same correlation distribution and family size as the large set of Kasami sequen ces [23 ]. The purp ose of this pap er is to study the w eigh t distribution of the co de C k in the non b inary case, namely we assume p is o d d, for a wide range of k that s atisfies gcd( n/ 2 , k ) = gcd( n/ 2 − k , 2 k ) = d b eing o d d . (1.2) Applying the tec hniqu es devel op ed in [1], we describ e some pr op erties of the ro ots to the equation δ p n − k y p n/ 2 − k +1 + γ y + δ = 0 Key wor ds and phr ases. Linear co de, weigh t distribut ion, exp onential sum, quadratic form. 1 2 XIANGYONG ZENG, NIAN LI, AND LEI HU with γ δ 6 = 0. Based on these prop erties and the theory of quadratic theory o v er finite fields of o dd c haracteristic, we complete ly determine the w eigh t distribution. As an app lication, these co des are also used to construct a class of nonbinary sequence families. The remainder of this pap er is organized as follo ws. Section 2 giv es some pr eliminaries and th e main result. S ection 3 considers the rank distribution of a class of quadratic forms. Section 4 determines the w eight distrib ution of the nonbinary co des, and these codes are u sed to construct a class of non binary sequence families w ith lo w correlation in Section 5. Section 6 concludes the study . 2. Preliminaries and Main R esul t F or p ositiv e inte gers n and l w ith l d ividing n , the trace f unction T r n l ( · ) from F p n to F p l is defined b y T r n l ( x ) = n/l − 1 P i =0 x p li , x ∈ F p n . F or the prop erties of the trace fu nction, please see [14]. Let q = p n . The field F q is an n -dimensional v ector space o v er F p . F or any g iven basis { α 1 , α 2 , · · · , α n } of F q o ve r F p , eac h elemen t x ∈ F q can b e uniquely represen ted as x = x 1 α 1 + x 2 α 2 + · · · + x n α n with x i ∈ F p for 1 ≤ i ≤ n . Under this repr esentati on, the field F q is iden tical to the F p -v ector sp ace F n p . A function f ( x ) on F p n is a quadr atic form if it can b e w ritten as a h omogeneous p olynomial of degree 2 on F n p , namely of the form f ( x 1 , · · · , x n ) = P 1 ≤ i ≤ j ≤ n a ij x i x j where a ij ∈ F p . The r ank of the quadr atic form f ( x ) is defined as the co dimension of th e F p -v ector space V f = n z ∈ F p n | f ( x + z ) = f ( x ) for all x ∈ F p n o , (2.1) denoted by rank( f ). Then | V f | = p n − rank( f ) . F or the qu ad r atic form f ( x ), there exists a sym metric matrix A suc h that f ( x ) = X T AX, where X T = ( x 1 , x 2 , · · · , x n ) ∈ F n p denotes the transp ose of a column v ector X . The determinant det( f ) of f ( x ) is defin ed to b e th e determinant of A , and f ( x ) is nonde gener ate if d et ( f ) 6 = 0. By Theorem 6.21 of [14], there exists a nonsingular matrix B such th at B T AB is a diagonal matrix. Making a n on s ingular linear su bstitution X = B Y with Y T = ( y 1 , y 2 , · · · , y n ), one has f ( x ) = Y T B T AB Y = n X i =1 a i y 2 i (2.2) for a 1 , a 2 , · · · , a n ∈ F p . Notice that a d egenerate quadratic form f ( x ) ov er F n p is p ossibly n ondegenerate o ve r F t p ( t < n ) after a nonsingular substitution. The quadr atic char acter of F q is defi n ed b y η ( x ) =    1 , if x is a square elemen t in F ∗ q , − 1 , if x is a n onsquare elemen t in F ∗ q , 0 , if x = 0 . The follo wing t wo lemmas about quadratic form will b e frequen tly used to pro ve the r esults of this pap er. A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 3 L emma 1 ( The or ems 6.26 and 6.27 of [14] ): F or o dd q , let f b e a non d egenerate quadr atic form o ve r F q in l indeterminates, and a f unction υ ( x ) o ve r F q defined by υ ( x ) =  − 1 , if x ∈ F ∗ q , q − 1 , otherwise . Then for ρ ∈ F q the n um b er of s olutions to the equ ation f ( x 1 , · · · , x n ) = ρ is q l − 1 + q l − 1 2 η  ( − 1) l − 1 2 ρ · det( f )  for o dd l , and q l − 1 + υ ( ρ ) q l − 2 2 η  ( − 1) l 2 det( f )  for ev en l . L emma 2 (The or em 5.15 of [14] ): Let ω b e a complex primitiv e p -th ro ot of u nit y . Then p − 1 X k =1 η ( k ) ω k = q ( − 1) p − 1 2 p. A p -ary [ m, l ] line ar c o de C is a linear subsp ace of F m p with dimension l . The Hamming weig ht of a co dew ord c 1 c 2 · · · c m of C is the num b er of nonzero c i for 1 ≤ i ≤ m . Let F b e a family of M p -ary sequences of p er io d p n − 1 giv en by F = n { s i ( t ) } p n − 2 t =0 | 0 ≤ i ≤ M − 1 o . The p erio dic c orr elatio n function of the sequences { s i ( t ) } and { s j ( t ) } in F is C i,j ( τ ) = p n − 2 X t =0 ω s i ( t ) − s j ( t + τ ) where 0 ≤ τ ≤ p n − 2. The t wo sequences { s i ( t ) } and { s j ( t ) } in F is cyclicly ine qu ivalent if | C i,j ( τ ) | < p n − 1 for any τ . Th e maximum magnitude C max of the correlation v alues is C max = max {| C i,j ( τ ) | : i 6 = j or τ 6 = 0 } . F rom now on, w e alw a ys assume that the pr ime p is o dd , and n = 2 m ≥ 4. Since T r n 1 ( δ x p k +1 ) = T r n 1 ( δ p n − k x p n − k +1 ) and δ p n − k runs through F p n as δ runs through F p n , without loss of generalit y , th e in teger k in the defin ition of code C k is assumed to satisfy 1 ≤ k < n/ 2. F or an o d d in teger t relativ ely p rime to m , th e inte ger k = m − t satisfies Equalit y (1.2), and d = 1. In particular, for t = 1, gcd( m, k ) = gcd( m − k , 2 k ) = 1. The parameter k = m − 1 corresp on d s to the binary Kasami co de [9], an d for this reason, we call these p -ary [ p n − 1 , 5 m ] linear co des C k with k satisfying Equalit y (1.2) the nonbinary Kasami c o des . The main resu lt of this p ap er is stated as the follo w ing theorem. The or em 1: F or an ev en in teger n = 2 m ≥ 4 and an y p ositiv e integ er k satisfying Equalit y (1.2), the weig ht distribu tion of the nonbinary Kasami co d es C k is giv en as T able 1. This theorem will b e prov en by the tec hn iqu es develo p ed in the n ext t wo sections. 3. Rank Distribution of Quad ra tic F or m Π γ ,δ ( x ) This section in v estigates the rank distrib ution of the quadr atic form Π γ ,δ ( x ) defin ed b y E q u alit y (1.1) for either γ 6 = 0 or δ 6 = 0. The p ossible rank v alues of Π γ ,δ ( x ) hav e a close relationship with the prop erties of the ro ots of the p olynomial g δ,γ ( y ) in the follo w ing Prop osition 1, which can b e pr o ve n b y applying the follo w ing lemma introdu ced b y Bluher [1]. 4 XIANGYONG ZENG, NIAN LI, AND LEI HU T able 1. W eig h t distribution of the nonbinary Kasami cod es C k w eight F requency 0 1 ( p − 1) p n − 1 ( p n − 1)(1 + p m + n − d − p m + n − 2 d + p m + n − 2 d − 1 + p m + n − 3 d − p n − 2 d ) ( p − 1)( p n − 1 − p n − 2 2 ) p d ( p m + 1)( p n − 1)( p n − 1 + ( p − 1) p n − 2 2 ) /  2( p d + 1)  ( p − 1)( p n − 1 + p n − 2 2 ) ( p n + d − 2 p n + p d )( p m − 1)( p n − 1 − ( p − 1) p n − 2 2 ) /  2( p d − 1)  ( p − 1) p n − 1 + p n − 2 2 p d ( p m + 1)( p n − 1)( p − 1)( p n − 1 − p n − 2 2 ) /  2( p d + 1)  ( p − 1) p n − 1 − p n − 2 2 ( p n + d − 2 p n + p d )( p m − 1)( p − 1)( p n − 1 + p n − 2 2 ) /  2( p d − 1)  ( p − 1) p n − 1 − p n + d − 1 2 p m − d ( p n − 1)( p − 1)( p n − d − 1 + p n − d − 1 2 ) / 2 ( p − 1) p n − 1 + p n + d − 1 2 p m − d ( p n − 1)( p − 1)( p n − d − 1 − p n − d − 1 2 ) / 2 ( p − 1)( p n − 1 + p n +2 d − 2 2 ) ( p m − d − 1)( p n − 1)  p n − 2 d − 1 − ( p − 1) p n − 2 d − 2 2  / ( p 2 d − 1) ( p − 1) p n − 1 − p n +2 d − 2 2 ( p m − d − 1)( p n − 1)( p − 1)( p n − 2 d − 1 + p n − 2 d − 2 2 ) / ( p 2 d − 1) L emma 3 (The or ems 5.4 and 5.6 of [1] ): Let h c ( x ) = x p s +1 − cx + c , c ∈ F ∗ p l . Then h c ( x ) = 0 has either 0, 1, 2, or p gcd( s,l ) + 1 ro ots in F ∗ p l . F urther, let N 1 denote the num b er of c ∈ F ∗ p l suc h that h c ( x ) = 0 has exactly one solution in F ∗ p l , then N 1 = p l − gcd( s,l ) and if x 0 ∈ F ∗ p l is the unique solution of the equation, then ( x 0 − 1) p l − 1 p gcd( s,l ) − 1 = 1. Pr op osition 1: Let g δ,γ ( y ) = δ p n − k y p m − k +1 + γ y + δ with γ δ 6 = 0, and d b e defined as in Equalit y (1.2). T h en (1) The equation g δ,γ ( y ) = 0 has either 0, 1, 2, or p d + 1 ro ots in F p n ; (2) If y 1 and y 2 are t wo different solutions of g δ,γ ( y ) = 0, then ( y 1 y 2 ) p n − 1 p d − 1 = 1; (3) If g δ,γ ( y ) = 0 has exactly one solution y 0 ∈ F p n , then y p n − 1 p d − 1 0 = 1. Pr o of: (1) Let y = − δ γ x and c = γ p m − k +1 δ p m − k ( p m +1) . Th e equation g δ,γ ( y ) = 0 is equiv alen t to x p m − k +1 − cx + c = 0 . (3.1) Since c ∈ F ∗ p m ⊆ F ∗ p n and gcd( m − k , n ) = gcd( m − k , 2 k ) = d by E qualit y (1.2), again b y Lemma 3, Equation (3.1) h as either 0, 1, 2, or p d + 1 r o ots in F p n . Thus, g δ,γ ( y ) = 0 also h as either 0, 1, 2, or p d + 1 ro ots in F p n . (2) Supp ose that y 1 , y 2 are t wo different solutions of g δ,γ ( y ) = 0. Th en y 1 y 2 ( y 1 − y 2 ) p m − k = y p m − k +1 1 y 2 − y p m − k +1 2 y 1 =  − ( γ y 1 + δ ) y 2 δ p n − k  −  − ( γ y 2 + δ ) y 1 δ p n − k  = δ ( y 1 − y 2 ) δ p n − k , i.e., y 1 y 2 = δ 1 − p n − k ( y 1 − y 2 ) 1 − p m − k . This together with Equality (1.2) imply ( y 1 y 2 ) p n − 1 p d − 1 = 1 . A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 5 (3) Supp ose that y 0 is the unique solution of g δ,γ ( y ) = 0. Sin ce y = − δ γ x , one has x 0 = − γ y 0 δ is the unique solution of Equation (3.1) in F p n . S ince c ∈ F p m , x p m 0 is also a solution of Equ ation (3.1). As a consequence, one has x 0 = x p m 0 , i.e., x 0 ∈ F p m . Th en, by Lemma 3, one has 1 = ( x 0 − 1) p m − 1 p d − 1 =  − γ y 0 δ − 1  p m − 1 p d − 1 =  − γ y 0 + δ δ  p m − 1 p d − 1 =  δ p n − k − 1 y p m − k +1 0  p m − 1 p d − 1 . (3.2) Similarly , if y is a solution of g δ,γ ( y ) = 0, one can verify th at y p m δ 1 − p m is also a solution of g δ,γ ( y ) = 0. Th us, y 0 = y p m 0 δ 1 − p m , i.e., y p m − 1 0 = δ p m − 1 . (3.3) Notice that d | m an d d | k b y Equ alit y (1.2). Th en, Equalities (3.2) and (3.3) imp ly 1 = δ ( p n − k − 1)( p m − 1) p d − 1 y ( p m − k +1)( p m − 1) p d − 1 0 = y ( p n − k − 1)( p m − 1) p d − 1 0 y ( p m − k +1)( p m − 1) p d − 1 0 = y p m − k p d − 1 ( p n − 1) 0 . This leads to y p n − 1 p d − 1 0 = y p m − k p d − 1 ( p n − 1) 0 ! p m + k = 1 .  R emark 1: (1) F or give n γ and δ with γ δ 6 = 0, by Prop osition 1(3), if g δ,γ ( y ) has exactly one solution in F p n , then the unique solution is ( p d − 1)-th p ow er in F p n . (2) By Proposition 1(2), if g δ,γ ( y ) h as at least t w o differen t solutio n s in F p n , th en all these solutions are ( p d − 1)-th p o wers in F p n , or none of them are ( p d − 1)-th p o wers in F p n . Ov er the fi nite fi eld of c haracteristic 2, an analogy of Pr op osition 1(2) w as obtained to study the cross correlation b et ween p erio d-d ifferen t m -sequences (Prop osition 2 of [8]). But the analogy of Prop osition 1(3) do es not exist in the characte ristic 2 case [8]. With Prop osition 1, the rank of the qu ad r atic form Π γ ,δ ( x ) can b e determined as follo ws. Pr op osition 2: F or γ 6 = 0 or δ 6 = 0, th e rank of Π γ ,δ ( x ) is either n , n − d , or n − 2 d . Pr o of: Th e integer p n − rank(Π γ ,δ ) is equal to the num b er of z ∈ F p n suc h that Π γ ,δ ( x + z ) = Π γ ,δ ( x ) (3.4) holds for all x ∈ F p n . Equ ation (3.4) holds if and only if T r m 1  γ ( x + z ) p m +1  + T r n 1  δ ( x + z ) p k +1  = T r m 1 ( γ x p m +1 ) + T r n 1 ( δ x p k +1 ) , or equiv alen tly if and only if T r m 1  γ ( xz p m + x p m z )  + T r n 1  δ ( xz p k + x p k z )  + T r m 1 ( γ z p m +1 ) + T r n 1 ( δ z p k +1 ) = 0 . By the equalit y xz p m + x p m z = T r n m ( xz p m ) and γ ∈ F p m , one has T r n 1 ( γ xz p m ) + T r n 1  δ ( xz p k + x p k z )  + T r m 1 ( γ z p m +1 ) + T r n 1 ( δ z p k +1 ) = 0 , 6 XIANGYONG ZENG, NIAN LI, AND LEI HU i.e., T r n 1   γ z p m + δ z p k + ( δ z ) p n − k  x  + T r m 1 ( γ z p m +1 ) + T r n 1 ( δ z p k +1 ) = 0 . Th us, Equation (3.4) holds for all x ∈ F p n if and only if γ z p m + δ z p k + ( δ z ) p n − k = 0 (3.5) and T r m 1 ( γ z p m +1 ) + T r n 1 ( δ z p k +1 ) = 0 . (3.6) By Equation (3.5), one has T r n 1 ( δ z p k +1 ) = − T r n 1 ( γ z p m +1 + δ p n − k z p n − k +1 ) = − T r n 1 ( γ z p m +1 ) − T r n 1 ( δ p n − k z p n − k +1 ) = − 2 T r m 1 ( γ z p m +1 ) − T r n 1 ( δ z p k +1 ) since γ z p m +1 ∈ F p m , namely , Equation (3.5) implies Equ ation (3.6). Then, it is sufficient to calculate the num b er of solutions to Equ ation (3.5). When δ = 0, one has γ 6 = 0 and then Equation (3.5) has one uniqu e s olution z = 0. In the sequel, w e only consider the case δ 6 = 0. If γ = 0, Equation (3.5) is equiv alen t to z ( z p 2 k − 1 + δ 1 − p k ) = 0. Th e n u m b er of all solutions to this equation is p gcd(2 k,n ) = p 2 d or 1 d ep endin g on whether − δ 1 − p k is a ( p 2 d − 1)-t h p o wer in F p n or not. Th us, in this case the r ank of Π γ ,δ ( x ) is either n − 2 d or n . F or γ 6 = 0, γ z p m + δ z p k + ( δ z ) p n − k = z p k ( δ + γ z p m − p k + δ p n − k z p n − k − p k ) = 0. Thus, w e only need consider the num b er of n on zero solutions to the equation δ + γ z p m − p k + δ p n − k z p n − k − p k = 0 . (3.7) Let y = z p k ( p m − k − 1) , then E quation (3.7) b ecomes g δ,γ ( y ) = δ p n − k y p m − k +1 + γ y + δ = 0 . By Pr op osition 1(1), g δ,γ ( y ) = 0 has either 0, 1, 2, or p d + 1 ro ots in F p n . Remark 1 and the fact gcd( p k ( p m − k − 1) , p n − 1) = p d − 1 show that Equation (3.7) has 0, p d − 1, 2( p d − 1), or ( p d − 1)( p d + 1) = p 2 d − 1 nonzero solutions in F p n . Then, Equ ation (3.5) has 1, p d , 2 p d − 1, or p 2 d solutions. Sin ce 2 p d − 1 is not a p o wer of p and hence is not the n u m b er of solutions of an F p -linearizatio n p olynomial, the n u m b er of solutions to Equ ation (3.4) is equal to 1, p d , or p 2 d . Therefore, the r ank of Π γ ,δ ( x ) is either n , n − d , or n − 2 d .  In order to fu rther determine the rank distr ib ution of Π γ ,δ ( x ), we define three sets R i = n ( γ , δ ) | rank(Π γ ,δ ) = n − i, ( γ , δ ) ∈ F p m × F p n \ { (0 , 0) } o (3.8) for i = 0, d , and 2 d . T o ac hieve this goal, we need to ev aluate th e exp onential sum S ( γ , δ, ǫ ) defined b y S ( γ , δ, ǫ ) = P x ∈ F p n ω Π γ ,δ ( x )+ T r n 1 ( ǫx ) (3.9) where γ ∈ F p m , δ, ǫ ∈ F p n . F or ρ ∈ F p , let N γ ,δ,ǫ ( ρ ) denote the num b er of solutions to the equation Π γ ,δ ( x ) + T r n 1 ( ǫx ) = ρ . Then, the exp onentia l sum can b e exp r essed as S ( γ , δ, ǫ ) = p − 1 X ρ =0 N γ ,δ,ǫ ( ρ ) ω ρ . (3.10) A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 7 Let f ( x ) = Π γ ,δ ( x ) b e as in Equalit y (1.1). F or conv enience, for i ∈ { 0 , d, 2 d } , w e defin e ∆ i = ( − 1) ⌊ n − i 2 ⌋ n − i Q j =1 a j , (3.11) where ⌊ n − i 2 ⌋ denotes the largest intege r not exceeding n − i 2 , and the co efficien ts a j are defined by Equalit y (2.2). In what follo ws, w e will study the v alues of S ( γ , δ , 0) according to the rank of Π γ ,δ ( x ), and then use them to determine the r ank distribution. Case 1: ( γ , δ ) ∈ R 0 . In this case, r ank(Π γ ,δ ) = n and every coefficient a i in Equalit y (2.2) is nonzero. Since det(Π γ ,δ )(det( B )) 2 = n Q i =1 a i , one has η (det(Π γ ,δ )) = η ( n Q i =1 a i ). Then by Lemma 1, N γ ,δ, 0 ( ρ ) = p n − 1 + v ( ρ ) p n − 2 2 η (∆ 0 ) and then by Equalit y (3.10), S ( γ , δ, 0) = η (∆ 0 ) p n 2 since p − 1 P ρ =0 v ( ρ ) ω ρ = p . Case 2: ( γ , δ ) ∈ R d . Since rank(Π γ ,δ ) = n − d , by Equ alit y (2.2), there exactly exist d in tegers i with 1 ≤ i ≤ n suc h that a i = 0. Without loss of generalit y , we assum e n − d Q i =1 a i 6 = 0 and a i = 0 for n − d < i ≤ n . T h en, Π γ ,δ ( x ) = n − d P i =1 a i y 2 i , and by Lemma 1, for o dd d , one has N γ ,δ, 0 ( ρ ) = p d  p n − d − 1 + p n − d − 1 2 η ( ρ ∆ d )  = p n − 1 + p n + d − 1 2 η ( ρ ∆ d ) . By Equalit y (3.10) and Lemma 2, S ( γ , δ, 0) = η (∆ d ) p n + d − 1 2 p − 1 P ρ =0 η ( ρ ) ω ρ = η (∆ d ) q ( − 1) p − 1 2 p n + d 2 . Case 3: ( γ , δ ) ∈ R 2 d . Similarly as in Case 2, we can assume n − 2 d Q i =1 a i 6 = 0 and a i = 0 for n − 2 d < i ≤ n . Then, a similar analysis shows that N γ ,δ, 0 ( ρ ) = p n − 1 + v ( ρ ) p n +2 d − 2 2 η (∆ 2 d ) and S ( γ , δ, 0) = η (∆ 2 d ) p n 2 + d . F or eac h i ∈ { 0 , d, 2 d } , we define t wo subsets of R i as R i,j = n ( γ , δ ) ∈ R i | η (∆ i ) = j o (3.12) for j = ± 1. 8 XIANGYONG ZENG, NIAN LI, AND LEI HU Since d is o dd, the cardin alit y of R d, 1 can b e pr o ve n to be the same as that of R d, − 1 in the follo wing lemma. L emma 4: | R d, 1 | = | R d, − 1 | . Pr o of: Let ( γ , δ ) ∈ R d and let u ∈ F ∗ p suc h th at its in verse elemen t satisfies η ( u − 1 ) = − 1. By Equalit y (3.9) and the analysis in Case 2, one has S ( uγ , uδ, 0) = P x ∈ F p n ω Π uγ ,uδ ( x ) = P x ∈ F p n ω u Π γ ,δ ( x ) = P ρ ∈ F p N γ ,δ, 0 ( u − 1 ρ ) w ρ = p n + d − 1 2 p − 1 P ρ =0 η ( u − 1 ρ ∆ d ) ω ρ = η ( u − 1 ) p n + d − 1 2 p − 1 P ρ =0 η ( ρ ∆ d ) ω ρ = − S ( γ , δ, 0) . The ab o ve equalit y shows that for j ∈ { 1 , − 1 } , and if ( γ , δ ) ∈ R d,j , then ( uγ , uδ ) ∈ R d, − j . T h u s, one has | R d, 1 | = | R d, − 1 | .  Applying Prop osition 1 and Lemma 4, the cardinalities of | R d | and | R d, ± 1 | can b e determined as b elo w. Pr op osition 3: T h e set R d consists of p m − d ( p n − 1) elements. F urther, | R d, ± 1 | = p m − d ( p n − 1) / 2. Pr o of: By Equalit y (3.8), it is su fficien t to determine the num b er of ( γ , δ ) ∈ F p m × F p n \ { (0 , 0) } suc h that Equ ation (3.5) has exactly p d solutions in F p n . By the pro of of Prop osition 2, this case can o ccur only if γ δ 6 = 0. Let W = { x p d − 1 | x ∈ F ∗ p n } b e the set of nonzero ( p d − 1)-th p o wers in F p n . By Prop osition 1(1), the equation g δ,γ ( y ) = 0 in P rop osition 2 h as either 0, 1, 2, or p d + 1 ro ots in F p n . When g δ,γ ( y ) = 0 has at least t wo different ro ots in F ∗ p n , by Prop osition 1(2) and Remark 1, if one of the solutions b elongs to W , then all these solutions are also in W and Equation (3.5) has at least 2( p d − 1) + 1 = 2 p d − 1 s olutions. If none of th ese solutions b elong to W , then Eq u ation (3.5) has only the un ique solution z = 0. Thus, in th is case the num b er of solutions to Equation (3.5) can n ev er b e p d . If g δ,γ ( y ) = 0 h as exactly one solution in F p n , by Prop osition 1(3) and Remark 1, Equation (3.5) has 1 × ( p d − 1) + 1 = p d solutions. Sin ce y = − δ γ x , g δ,γ ( y ) = 0 has exactly one solution in F ∗ p n if and only if Equ ation (3.1) has exactly one s olution in F ∗ p n . F urthermore, in th is case the un ique solution to Equation (3.1) b elongs to F ∗ p m b y the analysis in the pro of Prop osition 1(3). Wh en Equation (3.1) has exactly one solution in F ∗ p m , it has exactly one solution in F ∗ p n since this equ ation has either 0, 1, 2, or p d + 1 ro ots in F ∗ p n and its solutions in F p n \ F p m o ccur in pairs. Th er efore, g δ,γ ( y ) = 0 has exactly one solution in F ∗ p n if and only if Equation (3.1) has exactly one solution in F ∗ p m . By L emm a 3, the num b er of c ∈ F ∗ p m suc h that Equation (3.1 ) has exactly one solution in F p m is p m − d . F or an y fixed γ ∈ F ∗ p m , c = γ p m − k +1 δ p m − k ( p m +1) runs through F ∗ p m exactly ( p m + 1) times when δ run s through F ∗ p n . Th erefore, when γ and δ r un throughout F ∗ p m and F ∗ p n , resp ectiv ely , there are exactly | R d | = ( p m − 1) p m − d ( p m + 1) = p m − d ( p n − 1) elemen ts in R d . Th is together with Lemma 4 fi n ish the pro of.  A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 9 The follo wing prop osition describ es the sums of i -th p ow ers of S ( γ , δ , 0) for 1 ≤ i ≤ 3. Pr op osition 4: (i) X γ ∈ F p m X δ ∈ F p n S ( γ , δ, 0) = p n + m . (ii) X γ ∈ F p m X δ ∈ F p n S ( γ , δ, 0) 2 =  p n + m , p ≡ 3 mo d 4 , (2 p n − 1) p n + m , p ≡ 1 mo d 4 . (iii) X γ ∈ F p m X δ ∈ F p n S ( γ , δ, 0) 3 = p n + m ( p n + d + p n − p d ) . Pr o of: The p ro of of (i) is trivial, and we only giv e the pr o of of (ii) and (iii). (ii) It is true that P γ ∈ F p m ,δ ∈ F p n S ( γ , δ, 0) 2 = P γ ∈ F p m ,δ ∈ F p n P x,y ∈ F p n ω T r m 1 ( γ ( x p m +1 + y p m +1 ) ) + T r n 1 “ δ ( x p k +1 + y p k +1 ) ” = P x,y ∈ F p n P γ ∈ F p m ω T r m 1 ( γ ( x p m +1 + y p m +1 ) ) P δ ∈ F p n ω T r n 1 “ δ ( x p k +1 + y p k +1 ) ” = p n + m | T 2 | where T 2 consists of all s olutions ( x, y ) ∈ F p n × F p n to the f ollo wing system of equations  x p m +1 + y p m +1 = 0 , x p k +1 + y p k +1 = 0 . (3.13) If xy = 0, then x = y = 0 b y Equation (3.13). If xy 6 = 0, again by Equation (3.13), one has ( x y ) p k ( p m − k − 1) = 1 whic h implies ( x y ) p m − k − 1 =  ( x y ) p k ( p m − k − 1)  p n − k = 1, that is to sa y , x y ∈ F p m − k and then x y ∈ F ∗ p n ∩ F ∗ p m − k = F ∗ p d since gcd( m − k , n ) = gcd( m − k , 2 k ) = d by Equalit y (1.2). Let x = ty for some t ∈ F ∗ p d , then Equation (3.13) b ecomes t 2 + 1 = 0 (3.14) since x p m +1 = y p m +1 ( t p d · m d ) t = y p m +1 t 2 and similarly x p k +1 = y p k +1 t 2 . F or p ≡ 3 mo d 4, − 1 is a non-squ are elemen t in F p d since d is o dd. Thus, Equation (3.14) has n o solutions and | T 2 | = 1. F or p ≡ 1 mo d 4, − 1 is a squ are elemen t in F p d and Equation (3.14) h as 2 solutions in F ∗ p d . Th en, | T 2 | = 1 + 2( p n − 1) = 2 p n − 1. (iii) Similar analysis as in (ii) shows X γ ∈ F p m ,δ ∈ F p n S ( γ , δ, 0) 3 = p n + m | T 3 | , 10 XIANGYONG ZENG, NIAN LI, AND LEI HU where T 3 consists of all s olutions ( x, y , z ) ∈ F p n × F p n × F p n to the follo wing sys tem of equations  x p m +1 + y p m +1 + z p m +1 = 0 , x p k +1 + y p k +1 + z p k +1 = 0 . (3.15) F or xy z = 0, then x = y = z = 0, or there are exactly t wo nonzero elemen ts in { x, y , z } . Thus, by Equation (3.13), in this case the num b er of solutions to Equ ation (3.15) is equ al to 3( | T 2 | − 1) + 1 = 3 | T 2 | − 2. F or xy z 6 = 0, the num b er of solutions to Equation (3.15) is ( p n − 1) multiples of that to  x p m +1 + y p m +1 + 1 = 0 , x p k +1 + y p k +1 + 1 = 0 (3.16) where x, y ∈ F ∗ p n . By Equ ation (3.16), one has    x ( p m +1)( p k +1) =  − ( y p k +1 + 1)  p m +1 = y ( p m +1)( p k +1) + y ( p k +1) p m + y p k +1 + 1 , x ( p m +1)( p k +1) =  − ( y p m +1 + 1)  p k +1 = y ( p k +1)( p m +1) + y ( p m +1) p k + y p m +1 + 1 . This implies y ( p k +1) p m + y p k +1 − y ( p m +1) p k − y p m +1 = ( y p m + k − y )( y p m − y p k ) = 0 . By Equalit y (1.2), one has gcd ( m + k , n ) = d . Then y ∈ F p m + k ∩ F p n = F p d , or y ∈ F p m ∩ F p k = F p d , i.e., y ∈ F p d . Similarly , one has x ∈ F p d . By the fact gcd( m, k ) = d and a similar analysis as in the deriv ation of Equ alit y (3.14), Equation (3.16) is equiv alen t to x 2 + y 2 + 1 = 0 , x, y ∈ F ∗ p d . (3.17) By Lemma 1, the num b er of all solutions ( x, y ) ∈ F p d × F p d to the equation x 2 + y 2 + 1 = 0 is p d + v ( − 1) η ( − 1), where η ( − 1) = 1 if − 1 is a squ are elemen t in F p d , an d − 1 otherwise. Notice that the solutions satisfying xy = 0 and x 2 + y 2 + 1 = 0 in F p d do not exist for p ≡ 3 mo d 4, and th ey are exactly (0 , ± α p n − 1 4 ) and ( ± α p n − 1 4 , 0) for p ≡ 1 mo d 4, where α is a primitiv e element of F p n . As a consequence, the num b er of all solutions to Equation (3.17) is giv en by  p d + 1 , p ≡ 3 mo d 4 , ( p d − 1) − 4 , p ≡ 1 mo d 4 . Th us, one has | T 3 | = 1 + ( p d + 1)( p n − 1) = p n + d + p n − p d for p ≡ 3 mo d 4 , and | T 3 | = 3(2 p n − 1) − 2 + ( p d − 5)( p n − 1) = p n + d + p n − p d for p ≡ 1 mo d 4. Therefore, | T 3 | = p n + d + p n − p d .  With the ab o ve preparations, the rank distrib u tion of Π δ,γ ( x ) can b e determined as follo ws. Since S (0 , 0 , 0) = p n , by Prop osition 4 and the v alues of S ( γ , δ, 0) corresp onding to the rank of Π γ ,δ ( x ), one h as                  p n 2 ( | R 0 , 1 | − | R 0 , − 1 | ) + p n 2 + d ( | R 2 d, 1 | − | R 2 d, − 1 | ) + p n = P γ ∈ F p m P δ ∈ F p n S ( γ , δ, 0) , p n ( | R 0 , 1 | + | R 0 , − 1 | ) + ( − 1) p − 1 2 p n + d | R d | + p n +2 d ( | R 2 d, 1 | + | R 2 d, − 1 | ) + p 2 n = P γ ∈ F p m P δ ∈ F p n S ( γ , δ, 0) 2 , p 3 n 2 ( | R 0 , 1 | − | R 0 , − 1 | ) + p 3 n 2 +3 d ( | R 2 d, 1 | − | R 2 d, − 1 | ) + p 3 n = P γ ∈ F p m P δ ∈ F p n S ( γ , δ, 0) 3 . A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 11 T able 2. Rank distribu tion of the quadratic f orm Π γ ,δ ( x ) Rank F requency n ( p m + n +2 d + p n + p m + d − p m + n − p m + n + d − p 2 d ) / ( p 2 d − 1) n − d p m − d ( p n − 1) n − 2 d ( p m − d − 1)( p n − 1) / ( p 2 d − 1) This together with the equalit y | R 0 , 1 | + | R 0 , − 1 | + | R d | + | R 2 d, 1 | + | R 2 d, − 1 | = p n + m − 1 as well as Prop osition 3 giv es            | R 0 , 1 | = p d ( p m +1)( p n − 1) 2( p d +1) , | R 0 , − 1 | = ( p n + d − 2 p n + p d )( p m − 1) 2( p d − 1) , | R 2 d, 1 | = 0 , | R 2 d, − 1 | = ( p m − d − 1)( p n − 1) p 2 d − 1 . (3.18) Therefore, we ha v e the follo wing result. Pr op osition 5: When ( γ , δ ) runs through F p m × F p n \ { (0 , 0) } , the rank d istribution of the quadratic form Π γ ,δ ( x ) is giv en b y T able 2. 4. Weight Distribution of The Nonbinar y K asami Codes This sect ion determines the w eight d istribution of the nonbinary Kasami co des C k . F urthermore, w e also giv e the distribu tion of S ( γ , δ, ǫ ), wh ic h will b e used to deriv e the correlation d istribution of the sequence families pr op osed in next section. Since the we igh t of the co dewo r d c ( γ , δ , ǫ ) is equal to p n − 1 − ( N γ ,δ,ǫ (0) − 1) = p n − N γ ,δ,ǫ (0), it is sufficien t to fi n d N γ ,δ,ǫ (0) for any giv en γ , δ, ǫ . Under the basis { α 1 , α 2 , · · · , α n } of F p n o ve r F p , let ǫ = n P i =1 ǫ i α i with ǫ i ∈ F p . Then, T r n 1 ( ǫx ) = Λ T C X where Λ T = ( ǫ 1 , ǫ 2 , · · · , ǫ n ) ∈ F n p and the m atrix C = ( T r n 1 ( α i α j )) 1 ≤ i,j ≤ n , w hic h is nonsin gular since { α 1 , α 2 , · · · , α n } is a basis of F p n o ve r F p . Making a nonsin gular subs titution X = B Y as in Section 2, one has Π γ ,δ ( x ) + T r n 1 ( ǫx ) = Y T B T AB Y + Λ T C B Y = n P i =1 a i y 2 i + n P i =1 b i y i (4.1) where Λ T C B = ( b 1 , b 2 , · · · , b n ). Then, for any ρ ∈ F p , Π γ ,δ ( x ) + T r n 1 ( ǫx ) = ρ if and only if n P i =1 a i y 2 i + n P i =1 b i y i = ρ. W e calculate th e v alues of N γ ,δ,ǫ ( ρ ) ( ρ ∈ F p ) and the exp onentia l sum S ( γ , δ, ǫ ) as follo ws: Case 1: ( γ , δ ) = (0 , 0). F or ǫ 6 = 0, since the fun ction T r n 1 ( ǫx ) is linear from F p n to F p , th e w eigh t of c ( γ , δ, ǫ ) is ( p − 1) p n − 1 , and then S (0 , 0 , ǫ ) =  0 , ǫ 6 = 0 , p n , ǫ = 0 . Case 2: ( γ , δ ) 6 = (0 , 0). 12 XIANGYONG ZENG, NIAN LI, AND LEI HU Case 2.1: ( γ , δ ) ∈ R 0 . A substitution y i = z i − b i 2 a i for 1 ≤ i ≤ n leads to n X i =1 ( a i y 2 i + b i y i ) = ρ ⇐ ⇒ n X i =1 a i z 2 i = λ γ ,δ,ǫ + ρ, where λ γ ,δ,ǫ = n P i =1 b 2 i 4 a i . Then, for an y ρ ∈ F p and giv en ( γ , δ ) ∈ R 0 , by Lemma 1, one has N γ ,δ,ǫ ( ρ ) = p n − 1 + v ( λ γ ,δ,ǫ + ρ ) p n − 2 2 η (∆ 0 ) . (4.2) When ǫ runs through F p n , ( b 1 , b 2 , · · · , b n ) runs th rough F n p since C B is nonsingu lar. Notice that λ γ ,δ,ǫ is a qu adratic form with n v ariables b i for 1 ≤ i ≤ n . Then, for an y giv en ( γ , δ ) ∈ R 0 , by Lemma 1, when ǫ run s through F p n , one has λ γ ,δ,ǫ = n P i =1 b 2 i 4 a i = ρ ′ o ccurs p n − 1 + v ( ρ ′ ) p n − 2 2 η (∆ 0 ) times (4.3) for eac h ρ ′ ∈ F p since η  1 4 n n Q i =1 a i  = η  n Q i =1 a i  . Thus, when ǫ r u ns throu gh F p n , by E q u alities (4.2) and (4.3), N γ ,δ,ǫ (0) = p n − 1 + ( p − 1) p n − 2 2 η (∆ 0 ) o ccurs p n − 1 + ( p − 1) p n − 2 2 η (∆ 0 ) times, and N γ ,δ,ǫ (0) = p n − 1 − p n − 2 2 η (∆ 0 ) o ccurs ( p − 1)  p n − 1 − p n − 2 2 η (∆ 0 )  times. By Equalities (3.10) and (4.2), one has S ( γ , δ, ǫ ) = P ρ ∈ F p  p n − 1 + v ( λ γ ,δ,ǫ + ρ ) p n − 2 2 η (∆ 0 )  ω ρ = η (∆ 0 ) p n − 2 2 P ρ ∈ F p v ( λ γ ,δ,ǫ + ρ ) ω ρ = η (∆ 0 ) p n − 2 2 ω − λ γ ,δ,ǫ P ρ ∈ F p v ( λ γ ,δ,ǫ + ρ ) ω ρ + λ γ ,δ,ǫ = η (∆ 0 ) p n 2 ω − λ γ ,δ,ǫ , where the fourth equal sign holds since P ρ ∈ F p v ( λ γ ,δ,ǫ + ρ ) ω ρ + λ γ ,δ,ǫ = p . Notice that v ( − λ γ ,δ,ǫ ) = v ( λ γ ,δ,ǫ ). By Equalit y (4.3), for give n ( γ , δ ) ∈ R 0 , when ǫ runs through F p n , S ( γ , δ, ǫ ) = η (∆ 0 ) p n 2 ω ρ o ccurs p n − 1 + v ( ρ ) p n − 2 2 η (∆ 0 ) times for eac h ρ ∈ F p . Case 2.2: ( γ , δ ) ∈ R d . In this case, one has Π γ ,δ ( x ) + T r n 1 ( ǫx ) = n − d X i =1 a i y 2 i + n X i =1 b i y i . If th ere exists some b i 6 = 0 for n − d < i ≤ n , then for any ρ ∈ F p , N γ ,δ,ǫ ( ρ ) = p n − 1 and b y Equalit y (3.10) , S ( γ , δ, ǫ ) = 0. F urther, for giv en ( γ , δ ) ∈ R d , wh en ǫ run s through F p n , there are exactly p n − p n − d c hoices for ǫ suc h that th ere is at least one b i 6 = 0 with n − d < i ≤ n s ince C B is nonsingular. A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 13 If b i = 0 for all n − d < i ≤ n , a similar analysis as in Case 2.1 s h o ws that n − d X i =1 ( a i y 2 i + b i y i ) = ρ ⇐ ⇒ n − d X i =1 a i z 2 i = λ γ ,δ,ǫ + ρ, where λ γ ,δ,ǫ = n − d P i =1 b 2 i 4 a i and z i = y i + b i 2 a i for 1 ≤ i ≤ n − d . T hen, for an y ρ ∈ F p and giv en ( γ , δ ) ∈ R d , b y Lemma 1, one has N γ ,δ,ǫ ( ρ ) = p d  p n − d − 1 + p n − d − 1 2 η (( λ γ ,δ,ǫ + ρ )∆ d )  = p n − 1 + p n + d − 1 2 η (( λ γ ,δ,ǫ + ρ )∆ d ) (4.4) since n − d is o dd. F or giv en ( γ , δ ) ∈ R d , by Lemma 1, when ( b 1 , b 2 , · · · , b n − d ) run s through F n − d p , on e has λ γ ,δ,ǫ = n − d P i =1 b 2 i 4 a i = ρ ′ o ccurs p n − d − 1 + η ( ρ ′ ) p n − d − 1 2 η (∆ d ) times (4.5) for eac h ρ ′ ∈ F p . Since there are p − 1 2 square and non -squ are elemen ts in F p ∗ , resp ectiv ely , η ( λ γ ,δ,ǫ ) = 0 o ccurs p n − d − 1 times, and ± 1 o ccur p − 1 2  p n − d − 1 ± p n − d − 1 2 η (∆ d )  times, resp ectiv ely . Th erefore, w hen ( b 1 , b 2 , · · · , b n − d ) ru n s through F n − d p , N γ ,δ,ǫ (0) = p n − 1 o ccurs p n − d − 1 times , and N γ ,δ,ǫ (0) = p n − 1 ± p n + d − 1 2 η (∆ d ) o ccurs p − 1 2  p n − d − 1 ± p n − d − 1 2 η (∆ d )  times. By Equalit y (4.4), on e has S ( γ , δ, ǫ ) = P ρ ∈ F p  p n − 1 + p n + d − 1 2 η (( λ γ ,δ,ǫ + ρ )∆ d )  ω ρ = η (∆ d ) p n + d − 1 2 P ρ ∈ F p η ( λ γ ,δ,ǫ + ρ ) ω ρ = η (∆ d ) p n + d − 1 2 ω − λ γ ,δ,ǫ P ρ ∈ F p η ( λ γ ,δ,ǫ + ρ ) ω ρ + λ γ ,δ,ǫ = η (∆ d ) p n + d 2 q ( − 1) p − 1 2 ω − λ γ ,δ,ǫ , where the four th equal sign h olds due to Lemma 2. By Equalit y (4.5), f or giv en ( γ , δ ) ∈ R d , wh en ( b 1 , b 2 , · · · , b n − d ) ru n s through F n − d p , S ( γ , δ, ǫ ) = η (∆ d ) p n + d 2 q ( − 1) p − 1 2 ω ρ o ccurs p n − d − 1 + η ( − ρ ) p n − d − 1 2 η (∆ d ) times for eac h ρ ∈ F p . Case 2.3: ( γ , δ ) ∈ R 2 d . In this case, one has Π γ ,δ ( x ) + T r n 1 ( ǫx ) = n − 2 d X i =1 a i y 2 i + n X i =1 b i y i . Similarly as in Case 2.2, if there exists some b i 6 = 0 with n − 2 d < i ≤ n , then N γ ,δ,ǫ ( ρ ) = p n − 1 for an y ρ ∈ F p , and S ( γ , δ , ǫ ) = 0. F urther, for give n ( γ , δ ) ∈ R 2 d , wh en ǫ runs through F p n , there are p n − p n − 2 d c hoices for ǫ such that there is at least one b i 6 = 0 with n − 2 d < i ≤ n . 14 XIANGYONG ZENG, NIAN LI, AND LEI HU If b i = 0 for all n − 2 d < i ≤ n , a similar analysis sho w s that for an y ρ ∈ F p and giv en ( γ , δ ) ∈ R 2 d , b y Lemma 1, one has N γ ,δ,ǫ ( ρ ) = p n − 1 + v ( λ γ ,δ,ǫ + ρ ) p n +2 d − 2 2 η (∆ 2 d ) , (4.6) where λ γ ,δ,ǫ = n − 2 d P i =1 b 2 i 4 a i . F or giv en ( γ , δ ) ∈ R 2 d , wh en ( b 1 , b 2 , · · · , b n − 2 d ) runs through F n − 2 d p , by Lemma 1, one h as λ γ ,δ,ǫ = n − 2 d P i =1 b 2 i 4 a i = ρ ′ o ccurs p n − 2 d − 1 + v ( ρ ′ ) p n − 2 d − 2 2 η (∆ 2 d ) times (4.7) for eac h ρ ′ ∈ F p . Thus, when ( b 1 , b 2 , · · · , b n − 2 d ) ru n s through F n − 2 d p , N γ ,δ,ǫ (0) = p n − 1 + ( p − 1) p n +2 d − 2 2 η (∆ 2 d ) o ccurs p n − 2 d − 1 + ( p − 1) p n − 2 d − 2 2 η (∆ 2 d ) times, and N γ ,δ,ǫ (0) = p n − 1 − p n +2 d − 2 2 η (∆ 2 d ) o ccurs ( p − 1)  p n − 2 d − 1 − p n − 2 d − 2 2 η (∆ 2 d )  times. By Equalit y (4.6), one has S ( γ , δ, ǫ ) = η (∆ 2 d ) p n 2 + d ω − λ γ ,δ,ǫ . By Equalit y (4.7), f or giv en ( γ , δ ) ∈ R 2 d , when ( b 1 , b 2 , · · · , b n − 2 d ) ru n s through F n − 2 d p , S ( γ , δ, ǫ ) = η (∆ 2 d ) p n 2 + d ω ρ o ccurs p n − 2 d − 1 + v ( ρ ) p n − 2 d − 2 2 η (∆ 2 d ) times for eac h ρ ∈ F p . F or i ∈ { 0 , d, 2 d } and j ∈ { 1 , − 1 } , sin ce η (∆ i ) = j for ( γ , δ ) ∈ R i,j , Theorem 1 can b e pro v en b y the ab o ve analysis, Equalit y (3.18), and Prop osition 3. Pr o of of The or em 1: W e only giv e the frequen cies of the co dewords with weig hts ( p − 1) p n − 1 , ( p − 1)( p n − 1 − p n − 2 2 ), and ( p − 1) p n − 1 + p n + d − 1 2 . The other cases can b e pr o ve n in a similar wa y . The wei ght of c ( γ , δ , ǫ ) is equal to ( p − 1) p n − 1 if and only if N γ ,δ,ǫ (0) = p n − 1 , wh ich o ccurs only in Cases 1, 2.2 and 2.3. The frequen cy is equal to p n − 1 +  ( p n − p n − d ) + p n − d − 1  | R d | + ( p n − p n − 2 d ) | R 2 d | = ( p n − 1)(1 + p m + n − d − p m + n − 2 d + p m + n − 2 d − 1 + p m + n − 3 d − p n − 2 d ) . The wei ght of c ( γ , δ , ǫ ) is ( p − 1)( p n − 1 − p n − 2 2 ) if and only if N γ ,δ,ǫ (0) = p n − 1 + ( p − 1) p n − 2 2 , wh ich o ccurs only in Case 2.1. The frequen cy is equal to ( p n − 1 + ( p − 1) p n − 2 2 ) | R 0 , 1 | = p d ( p m + 1)( p n − 1)( p n − 1 + ( p − 1) p n − 2 2 ) /  2( p d + 1)  . F or c ( γ , δ , ǫ ), its wei gh t equals to ( p − 1) p n − 1 + p n + d − 1 2 if and only if N γ ,δ,ǫ (0) = p n − 1 − p n + d − 1 2 , whic h o ccurs only in Case 2.2. The fr equency is equal to p − 1 2 ( p n − d − 1 + p n − d − 1 2 ( − 1)) | R d, − 1 | + p − 1 2 ( p n − d − 1 − p n − d − 1 2 ) | R d, 1 | = p m − d ( p n − 1)( p − 1)( p n − d − 1 − p n − d − 1 2 ) / 2 .  R emark 2: (1) F rom T able 1, the co de C k has 9 different w eights for d = 1, and 10 differen t w eights for d > 1. (2) The codewords with w eight ( p − 1)( p n − 1 − p n +2 d − 2 2 ) or ( p − 1) p n − 1 + p n +2 d − 2 2 do not exist in C k since | R 2 d, 1 | = 0. The follo wing result can also b e similarly pr o ve n and we omit its pro of here. A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 15 T able 3. Distribution of S ( γ , δ , ǫ ) (w ith ρ v arying in F p ) S ( γ , δ, ǫ ) F requency p n 1 0 ( p n − 1)  1 + p m + n − d − p m + n − 2 d + p m + n − 3 d − p n − 2 d  p n 2 ω ρ p d ( p m + 1)( p n − 1)  p n − 1 + v ( ρ ) p n − 2 2  /  2( p d + 1)  − p n 2 ω ρ ( p n + d − 2 p n + p d )( p m − 1)  p n − 1 − v ( ρ ) p n − 2 2  /  2( p d − 1)  p n + d 2 q ( − 1) p − 1 2 ω ρ p m − d ( p n − 1)( p n − d − 1 + η ( − ρ ) p n − d − 1 2 ) / 2 − p n + d 2 q ( − 1) p − 1 2 ω ρ p m − d ( p n − 1)( p n − d − 1 − η ( − ρ ) p n − d − 1 2 ) / 2 − p n 2 + d ω ρ ( p m − d − 1)( p n − 1)  p n − 2 d − 1 − v ( ρ ) p n − 2 d − 2 2  / ( p 2 d − 1) Pr op osition 6: F or n = 2 m ≥ 4, the exp onen tial sum S ( γ , δ, ǫ ) defined in Equ alit y (3.9) has the distribution give n in T able 3 when ( γ , δ, ǫ ) run s through F p m × F p n × F p n . 5. A Class of Non binar y Seque nce F amilies By choosing cyclicly inequiv alen t cod ew ords from C k , a family of non binary sequences is defined by F k = n { s a,b ( α t ) } 0 ≤ t ≤ p n − 2 | a ∈ F p m , b ∈ F p n o , (5.1) where α is a pr imitiv e elemen t of F p n , and s a,b ( α t ) = T r m 1 ( aα ( p m +1) t ) + T r n 1 ( bα ( p k +1) t + α t ) . The subfamily consisting of the sequences { s a, 0 ( α t ) } 0 ≤ t ≤ p n − 2 for all a ∈ F p m (and with fixing b = 0) has b een considered in [13] and App endix A of [10] resp ectiv ely , and its correlat ion distribu tion w as determined in [13]. F or k = m + 1, p ossib le correlation v alues of the family F m +1 w as discus s ed in [22], but the correlation distr ib ution remains un solv ed. T o aim at the correlation distribution of F k , w e write the correlation function b et we en t wo sequ en ces s a 1 ,b 1 and s a 2 ,b 2 as C a 1 b 1 ,a 2 b 2 ( τ ) = p n − 2 P t =0 ω s a 1 ,b 1 ( t ) − s a 2 ,b 2 ( t + τ ) = p n − 2 P t =0 ω T r m 1 ( ( a 1 − a 2 α ( p m +1) τ ) α ( p m +1) t ) + T r n 1 “ ( b 1 − b 2 α ( p k +1) τ ) α ( p k +1) t +(1 − α τ ) α t ” = − 1 + P x ∈ F p n ω T r m 1 ( ( a 1 − a 2 α ( p m +1) τ ) x p m +1 ) + T r n 1 “ ( b 1 − b 2 α ( p k +1) τ ) x p k +1 +(1 − α τ ) x ” = − 1 + S ( λ 1 , λ 2 , λ 3 ) , where λ 1 = a 1 − a 2 α ( p m +1) τ , λ 2 = b 1 − b 2 α ( p k +1) τ , λ 3 = 1 − α τ . (5.2) With this, the distr ib ution of correlation v alues of th e f amily F k can b e describ ed in terms of the exp onenti al sum S ( λ 1 , λ 2 , λ 3 ). A simple prop ert y of S ( γ , δ, ǫ ) is d escrib ed as b elo w. L emma 5: F o r an y giv en ǫ ∈ F ∗ p n , when ( γ , δ ) ru ns through F p m × F p n , the distribu tion of S ( γ , δ, ǫ ) is the same as that of S ( γ , δ, 1). 16 XIANGYONG ZENG, NIAN LI, AND LEI HU T able 4. Correlation distribu tion of the sequence f amily F k (with ρ v arying in F ∗ p ) Correlation v alue F requency p n − 1 p 3 n 2 − 1 p 3 n 2 ( p n − 2)  1 + p 3 n 2 − d − p 3 n 2 − 2 d + p 3 n 2 − 3 d − p n − 2 d  p n 2 − 1 p 3 n 2 + d ( p n 2 + 1)  ( p n − 2)( p n − 1 + p n 2 − p n − 2 2 ) + 1  /  2( p d + 1)  p n 2 ω ρ − 1 p 3 n 2 + d ( p n 2 + 1)( p n − 2)( p n − 1 − p n − 2 2 ) /  2( p d + 1)  − p n 2 − 1 p 3 n 2 ( p n + d − 2 p n + p d )  ( p n − 2)( p n − 1 − p n 2 + p n − 2 2 ) + 1  /  2( p n 2 + 1)( p d − 1)  − p n 2 ω ρ − 1 p 2 n − 1 ( p n + d − 2 p n + p d )( p n − 2) /  2( p d − 1)  ± p n + d 2 q ( − 1) p − 1 2 − 1 p 2 n − d  ( p n − 2) p n − d − 1 + 1  / 2 p n + d 2 q ( − 1) p − 1 2 ω ρ − 1 p 2 n − d ( p n − 2)  p n − d − 1 + η ( − ρ ) p n − d − 1 2  / 2 − p n + d 2 q ( − 1) p − 1 2 ω ρ − 1 p 2 n − d ( p n − 2)  p n − d − 1 − η ( − ρ ) p n − d − 1 2  / 2 − p n 2 + d − 1 p 3 n 2 ( p n 2 − d − 1)  ( p n − 2)( p n − 2 d − 1 − p n 2 − d + p n 2 − d − 1 ) + 1  / ( p 2 d − 1) − p n 2 + d ω ρ − 1 p 3 n 2 ( p n 2 − d − 1)( p n − 2)( p n − 2 d − 1 + p n 2 − d − 1 ) / ( p 2 d − 1) Pr o of: F or an y fixed ǫ ∈ F ∗ p n , one has S ( γ , δ, ǫ ) = P x ∈ F p n ω T r m 1 ( γ x p m +1 )+ T r n 1 ( δx p k +1 + ǫx ) = P y ∈ F p n ω T r m 1 ( γ ǫ − ( p m +1) y p m +1 )+ T r n 1 ( δǫ − ( p k +1) y p k +1 + y ) = S ( γ ǫ − ( p m +1) , δ ǫ − ( p k +1) , 1) . F or an y fi xed ǫ ∈ F ∗ p n , when ( γ , δ ) runs through F p m × F p n , so do es ( γ ǫ − ( p m +1) , δ ǫ − ( p k +1) ). Thus, the distribution of S ( γ , δ , ǫ ) is the same as that of S ( γ , δ, 1).  The or em 2: Let F k b e the family of sequences d efined in Equalit y (5.1). Th en, F k is a family of p 3 n 2 non b inary sequences with p erio d p n − 1, and its maxim um co r relation magnitude is equal to p n 2 + d + 1. F urther, its correlation distr ibution is giv en in T able 4. Pr o of: By Equ alit y (5.2), for any fixed ( a 2 , b 2 ) ∈ F p m × F p n , wh en ( a 1 , b 1 ) runs through F p m × F p n and τ v aries fr om 0 to p n − 2, ( λ 1 , λ 2 , λ 3 ) runs through F p m × F p n ×{ F p n \{ 1 }} one time. T h u s, th e correlation distribution of F k is p 3 n 2 times as that of S ( γ , δ, ǫ ) − 1 when ( γ , δ, ǫ ) ru n s through F p m × F p n × { F p n \ { 1 }} . By Prop osition 3, E qualit y (3.18) and the p ossible v alues of S ( γ , δ, 0) corresp onding to ( γ , δ ), the distribution of S ( γ , δ , 0) − 1 is obtained when ( γ , δ ) run s through F p m × F p n . This together with Prop osition 6 give the distrib ution of S ( γ , δ, γ ) − 1 as ( γ , δ, ǫ ) runs through F p m × F p n × F ∗ p n . By L emma 5, the distribu tion of S ( γ , δ, ǫ ) − 1 can b e d etermin ed when ( γ , δ , ǫ ) run s through F p m × F p n × { F ∗ p n \ { 1 }} . T ogether with the distribution of S ( γ , δ, 0) − 1 as ( γ , δ ) runs through F p m × F p n , the correlation distribution is given as T able 4. By the defi n ition of the sequ en ce { s a,b ( α t ) } 0 ≤ t ≤ p n − 2 , it is easy to chec k its p erio d is p n − 1. F rom the correlation distrib ution give n in T able 4, one easily knows there are exactly p 3 n 2 sequences in th e family F k and the maxim um magnitude is p n 2 + d + 1. This finishes the pro of.  A CLASS OF NONBINAR Y CODES AND THEIR WEIGHT DISTRIBUTION 17 R emark 3: (1) By T able 4, the correlation f unction of F k tak es 5 p + 2 v alues. T ak e k = m − t for an y o d d in tegers t relativ ely prim e to m as stated in Section 2, th en d = 1 and the f amilies F k ha ve the maximum magnitude p n 2 +1 + 1. (2) Notice that if we remo ve the term T r n 1 ( α t ) in Eq u alit y (5.1) and d efine a sequen ce set as n { T r m 1 ( aα ( p m +1) t ) + T r n 1 ( bα ( p k +1) t ) } 0 ≤ t ≤ p n − 2 | a ∈ F p m , b ∈ F p n o , the p erio d of eac h sequence in that set is n ot larger than ( p n − 1) / 2. The sequence families prop osed in the p resen t pap er do not con tain any sequences in that s et. 6. Conclud ing Remar ks F or an ev en inte ger n = 2 m ≥ 4 and any p ositive in teger k satisfying E qualit y (1.2), a class of binary Kasami co d es has b een extended to o dd p -ary case. Th ese p -ary [ p n − 1 , 5 n 2 ] co des C k can ac hiev e the maximal v alue ( p − 1) p n − 1 − p n 2 for their minim um distances by taking d = 1, and in this case, a family F k of p 3 n 2 p -ary sequences with p erio d p n − 1 can b e defined and hav e the maximum correlation magnitude p n 2 +1 + 1. W e h ad tried to remo v e th e restriction in E q u alit y (1.2) on the parameters m and k . C learly , the assumption of Equ ality (1.2) is equiv alen t to sa y that d = gcd( m, k ) is o d d and m/d and k /d hav e differen t parit y . If d is eve n , Prop osition 2 still h olds and hence th e rank of Π γ ,δ is still n , n − d , or n − 2 d . If b oth m/d and k /d are o d d, then the rank of Π γ ,δ can b e sho w n to b e n , n − 2 d , or n − 4 d . 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The ory, vol. 43, n o. 4, pp. 1354-1360, July 1997. [21] J. Y uan, C. Carlet, and C. Ding, “The w eight distribution of a class of linear co des from p erfect nonlinear functions,” IEEE T r ans. Inform. The ory, vol. 52, no. 2, pp . 712-717, F eb. 2006. [22] Y. Xia, X. Zeng, and L. Hu , “The large set of p -ary Kasami sequen ces,” preprint. [23] X. Zeng, J. Q. Liu, and L. Hu, “Generalized Kasami sequen ces: th e la rge set,” IEEE T r ans. Inform. The ory , vol. 53, n o. 7, pp. 2587-2598, July 2007. Xiangyong Zeng and N ian Li are with the F a cul ty of Ma them a tics a nd Computer Science, Hubei University, Wuhan, C hina. Email: xzeng@hubu.edu.cn Lei Hu i s with the St a te Ke y Labora tor y of Informa tion Securi ty, Gradua te Un iversity of Chinese Academy of Sciences, Beijing, Chi na. E m ail: hu@is.ac.cn