On Binary Cyclic Codes with Five Nonzero Weights

On Binary Cyclic Co des with Fiv e Nonzero W eigh ts No vem b er 9, 2018 Jinquan Luo 1 1 J.Luo is with the School of Mathematics, Y angzhou Universit y , Jiangsu Province, 225009, China. and also with the Division of Mathematica l Sciences, Schoo l of Physics and Ma themat- ical Sciences, Na ny ang T ec hnolog ical Univ ersity , Singap ore. This w or k is suppor ted by NRF Comp etitive Resear ch Prog ram NRF-CRP 2-200 7-03, Sing ap ore. 1 Abstract Let q = 2 n , 0 ≤ k ≤ n − 1, n/ gcd( n, k ) b e o dd and k 6 = n/ 3 , 2 n/ 3. In this pap er the v alue distribution of follow ing exp o nential sums X x ∈ F q ( − 1) T r n 1 ( αx 2 2 k +1 + β x 2 k +1 + γ x ) ( α, β , γ ∈ F q ) is determined. As an application, the w eight distribution of the binary cyclic co de C , with parity-c hec k p olynomial h 1 ( x ) h 2 ( x ) h 3 ( x ) where h 1 ( x ), h 2 ( x ) and h 3 ( x ) a re the minimal p olynomials of π − 1 , π − (2 k +1) and π − (2 2 k +1) resp ectiv ely for a primitiv e elemen t π of F q , is also determined. Index terms: Exp onen tial sum, Cyclic co de, Moment iden tity , W eigh t distri- bution, Sequence 2 1 In tro ductio n Basic knowledge on finite fields could b e found in [16]. The following notations are fixed throughout this pap er except for sp ecific statemen ts. • Let n b e a p ositiv e intege r, q = 2 n , F q b e the finite field of order q . Let π b e a primitive eleme nt of F q . • Let T r j i : F 2 i → F 2 j b e the trace mapping, and χ ( x ) = ( − 1) T r n 1 ( x ) b e the canonical additive c haracter on F q . • Let k b e a p ositiv e integer, 1 ≤ k ≤ n − 1 and k / ∈ { n 3 , 2 n 3 } . Let d = gcd( n, k ), q 0 = 2 d , and s = n/d . Assume s is o dd. F or cyclic code C with length l , let A i b e the n umber of co dew ords in C with Hamming we ight i . The we ight distribution { A 0 , A 1 , · · · , A l } is an imp ort a n t researc h ob ject for b oth theoretical and application inte rests in co ding theory . Classical co ding theory reve als tha t the w eigh t o f eac h co dew ord can b e expressed b y exp onential sums so that the w eight distribution of C can b e determined if the corresp onding exp onential sums can b e calculated explicitly (see F eng and Luo[8], Kasami[15], Luo and F eng[17]-[18], Moisio[21], v an der Vlugt[27], W olfmann[2 8 ], Zeng, Liu and Hu[31]). More precise sp eaking, let q = 2 n , C b e the binary cyclic co de with length l = q − 1 and pa rit y-c hec k p olynomial h ( x ) = h 1 ( x ) · · · h u ( x ) ( u ≥ 1) where h i ( x ) (1 ≤ i ≤ u ) are distinct irreducible p o lynomials in F 2 [ x ] with the same degree e i (1 ≤ i ≤ u ), then dim F 2 C = u P i =1 e i . Let π − s i b e a zero of h i ( x ), 1 ≤ s i ≤ q − 2 (1 ≤ i ≤ u ) . Then the codewords in C can b e expressed by c ( α 1 , · · · , α u ) = ( c 0 , c 1 , · · · , c l − 1 ) ( α 1 , · · · , α u ∈ F q ) where c i = u P λ =1 T r n 1 ( α λ π is λ ) (0 ≤ i ≤ l − 1). Therefore the Hamming weigh t of the co dew ord c = c ( α 1 , · · · , α u ) is w H ( c ) = 2 n − 1 − 1 2 S ( α 1 , · · · , α u ) (1) 3 where f ( x ) = α 1 x s 1 + α 2 x s 2 + · · · + α u x s u ∈ F q [ x ], F ∗ q = F q \{ 0 } , and S ( α 1 , · · · , α u ) = X x ∈ F q ( − 1) T r n 1 ( α 1 x s 1 + ··· + α u x s u ) . In this w a y , the w eight distribution of cyclic co de C can b e deriv ed from the explicit ev aluat ing of the exp onen tial sums S ( α 1 , · · · , α u ) ( α 1 , · · · , α u ∈ F q ) . Let h 1 ( x ), h 2 ( x ) and h 3 ( x ) b e the minimal p olynomials of π − 1 , π − (2 k +1) and π − (2 2 k +1) o v er F 2 resp ectiv ely . Then deg h i ( x ) = n f or i = 1 , 2 , 3 . Let C b e the binary cyclic co des with length l = q − 1 and parit y-ch ec k p olynomials h 1 ( x ) h 2 ( x ) h 3 ( x ). It is a consequence that C is the dual of the binary cyclic co de whose defining zero es are π 2 2 k +1 , π 2 k +1 and π . Then the dimensions of C is 3 n b y excluding sev eral special cases: k = n/ 3 , 2 n/ 3. F or α, β , γ ∈ F q , define the expo nen tial sum S ( α , β , γ ) = X x ∈ F q χ  αx 2 2 k +1 + β x 2 k +1 + γ x  . (2) Then the complete w eight distributions of C can b e deriv ed f rom the explicit ev aluatio n of S ( α, β , γ ). F or k = 1, C ⊥ or its extended co de, is the we ll-known triple-error correcting BCH co de whic h has b een extensiv ely studied. F or instance, (1). The w eigh t distribution of C has been calculated, see MacWilliams a nd Sloane[20], pp.669, K a sami[15] for n o dd and Berlek amp[2 ] for n ev en. (2). The cov ering radius of C is 5, whic h has b een prov en in Assm us and Mattson[1]. (3). The coset distribution of C ⊥ has b een determined, see Charpin, Helleseth and Zinov iev[3]. (4). The we ight of coset leaders to the extended co de of C ⊥ has b een studied, see Charpin, Helleseth and Zinoviev[4]-[5] and Charpin, Zino viev[6 ]. Let a = ( a λ ) 2 n − 2 λ =0 and b = ( b λ ) 2 n − 2 λ =0 b e t w o m -seque nces with p erio d q − 2. The c orr elation function o f a and b for a shift τ is defined by M a,b ( τ ) = 2 n − 2 X λ =0 ( − 1) a ( λ ) − b ( λ + τ ) (0 ≤ τ ≤ q − 2) . 4 Binary sequences with low cross correlation and a ut o -correlation ar e widely used in Co de Division Multiple Access(CDMA) spread sp ectrum (see Simon, Om ura and Sc holtz[26]). Pairs of binary m -sequence s with few-v alued correlations ha v e b een extensiv ely studied for sev eral decades, se e Dobb ertin, F elk e, Helleseth and Rosendahl[7], Helleseth[9 ], Helleseth, Kholo sha and Ness[10], Helleseth a nd Kumar [11], Hu, Zeng, Li and Jiang[12], Johansen, Helleseth and T ang[14], Ness and Helleseth[23]- [22], Niho[24], Rosendahl[25 ], Y u and Go ng[29]-[30] and refer- ences therein. Recen tly , the exp onential sum S ( α, β , 0) with n/d o dd has b een studied, in terms of certain combination of exp onen tial sums, see Johansen and Helleseth[13]. In particular, for the case k = 1 and n o dd, the fiv e-v alued corre- lation distribution b etw een t w o m -sequenc es has b een determined. Based on the exp onen tial sum S ( α, β , γ ), w e could define a family of m - sequence s F = F 1 ∪ F 2 with F 1 =   T r n 1 ( απ λ (2 2 k +1) + β π λ (2 k +1) + π λ )  q − 2 λ =0    α, β ∈ F q  and F 2 =   T r n 1 ( απ λ (2 2 k +1) + π λ (2 k +1) )  q − 2 λ =0    β ∈ F q  [   T r n 1 ( π λ (2 2 k +1) )  q − 2 λ =0  . The seque nce family F has family size 2 2 n + 2 n + 1. In this pap er, we will f o cus on the case n/d is o dd and it is presen ted a s follows . In section 2 w e in tro duce some preliminaries. In section 3 w e will determine the v alue distribution of S ( α , β , γ ), and at the same time, the w eigh t distribution of C . As a corollary , the p ossible cor r elation v alues among the sequences in F can b e obtained. But unfort una t ely , we could not determine the correlation distribution. The main tec hniques w e will emplo y are binary quadratic form theory and the third-p ow er momen t iden tities of S ( α, β , γ ). 2 Preliminaries W e follow the notations in section 1. The first tec hnique is quadratic form theory o v er F q 0 . Let H b e an s × s matrix ov er F q 0 . F or the quadratic form F : F s q 0 → F q 0 , F ( x ) = X H X T ( X = ( x 1 , · · · , x s ) ∈ F s q 0 ) , (3) 5 define r F of F to b e the rank of the sk ew-symmetric matrix H + H T . Then r F is ev en. W e hav e the follo wing result on the exp onential sum of binary quadratic forms (see [19]). L emma 1. F o r the quadr atic form F = X H X T define d in (3), X X ∈ F s q 0 ( − 1) T r d 1 ( F ( X )) = ± q s − r F 2 0 or 0 Mor e over, if r F = s , then X X ∈ F s q 0 ( − 1) T r d 1 ( F ( X )) = ± q s 2 0 The follo wing result will b e used in the study of S ( α, β , γ ) (see [27]). L emma 2. F o r the fixe d quadr atic form defi n e d in (3), the v alue distribution of P X ∈ F s q 0 ( − 1) T r d 1 ( F ( X )+ AX T ) when A runs thr ough F s q 0 is shown as fol lowing v al ue mul tiplicity 0 q s 0 − q r F 0 q s − r F 2 0 1 2 ( q r F 0 + q r F 2 0 ) − q s − r F 2 0 1 2 ( q r F 0 − q r F 2 0 ) Note that the field F q is a vector space ov er F q 0 with dimension s .F or fixed basis v 1 , · · · , v s of F q o v er F q 0 , eac h x ∈ F q can b e uniquely expressed a s x = x 1 v 1 + · · · + x s v s ( x i ∈ F q 0 ) . Th us w e hav e the follow ing F q 0 -linear isomorphism: F q ∼ − → F s q 0 , x = x 1 v 1 + · · · + x s v s 7→ X = ( x 1 , · · · , x s ) . With this isomorphism, a function f : F q → F q 0 induces a f unction F : F s q 0 → F q 0 where fo r X = ( x 1 , · · · , x s ) ∈ F s q 0 , F ( X ) = f ( x ) with x = x 1 v 1 + · · · + x s v s . In this w a y , function f ( x ) = T r n d ( γ x ) for γ ∈ F q induces a linear form F ( X ) = T r n d ( γ x ) = s X i =1 T r n d ( γ v i ) x i = A γ X T (4) 6 where A γ = (T r n d ( γ v 1 ) , · · · , T r n d ( γ v s )) , and f α,β ( x ) = T r n d ( αx p 2 k +1 + β x p k +1 ) for α, β ∈ F q induces a quadratic form F α,β ( X ) = X H α,β X T F rom Lemma 1, for α, β , γ ∈ F q , in order to determine the v alues of S ( α , β , γ ) = X x ∈ F q ( − 1) T r n 1 ( αx 2 2 k +1 + β x 2 k +1 + γ x ) = X X ∈ F s q 0 ( − 1) T r d 1 ( X H α,β X T + A γ X T ) , w e need to determine the rank of H α,β + H T α,β o v er F q 0 . L emma 3. F or α, β ∈ F q and ( α, β ) 6 = (0 , 0) , let r α,β b e the r ank of H α,β + H T α,β . Then the p ossible values of r α,β ar e s − 1 and s − 3 . Pr o of. F or Y = ( y 1 , · · · , y s ) ∈ F s q 0 , y = y 1 v 1 + · · · + y s v s ∈ F q , w e kno w t ha t F α,β ( X + Y ) − F α,β ( X ) − F α,β ( Y ) = 2 X H α,β Y T (5) is equal to f α,β ( x + y ) − f α,β ( x ) − f α,β ( y ) = T r n d  y 2 2 k ( α 2 2 k x 2 4 k + β 2 2 k x 2 3 k + β 2 k x 2 k + α x )  (6) Let φ α,β ( x ) = α 2 2 k x 2 4 k + β 2 2 k x 2 3 k + β 2 k x 2 k + αx. (7) Therefore, r α,β = r ⇔ the n um b er of common solutions of X H α,β Y T = 0 for all Y ∈ F s q 0 is q s − r 0 , ⇔ the n umber of common solutions of T r n d  y 2 2 k · φ α,β ( x )  = 0 for all y ∈ F q is q s − r 0 , ⇔ φ α,β ( x ) = 0 has q s − r 0 solutions in F q . F or a fixed algebraic closure F 2 ∞ of F 2 , since the degree o f 2 2 k -linearized p olynomial φ α,β ( x ) is 2 4 k and φ α,β ( x ) = 0 has no m ultiple ro ots in F 2 ∞ (this fact follo ws from φ ′ α,β ( x ) = α ∈ F ∗ q ), then the zero es of φ α,β ( x ) in F 2 ∞ , sa y V , form an F 2 k - v ector space of dimension 4. Then V ∩ F 2 n is a vector space on F 2 gcd( n,k ) = F 2 d of dimension less that or equal to 4 since an y elemen ts in F 2 n whic h are linear indep enden t o v er F 2 d are a lso linear indep enden t ov er F 2 k (see [ ? ], Lemma 4). Note that s is o dd a nd r α,β is alw ays ev en. Hence the p ossible v a lues of r α,β are s − 1 and s − 3. Another tec hnique to determine the v alue distribution of S ( α , β , γ ) is the third-p ow er momen t iden tity of S ( α, β , γ ). 7 L emma 4. F o r the exp onential sum and S ( α , β , γ ) , we have X α,β ,γ ∈ F q S ( α , β , γ ) 3 = (2 n + d + 2 n − 2 d ) · 2 3 n . Pr o of. W e can calculate P α,β ∈ F q S ( α , β , γ ) 3 = P x,y ,z ∈ F q P α ∈ F q χ  α  x 2 2 k +1 + y 2 2 k +1 + z 2 2 k +1  P β ∈ F q χ  β  x 2 k +1 + y 2 k +1 + z 2 k +1  P γ ∈ F q χ ( γ ( x + y + z )) = M 3 · 2 3 n where M 3 is the n um b er of solutions to the system of equations        x + y + z = 0 x 2 k +1 + y 2 k +1 + z 2 k +1 = 0 x 2 2 k +1 + y 2 2 k +1 + z 2 2 k +1 = 0 (8) • If xy z = 0, w e ma y assume x = 0 and then y = z whic h gives 2 n solutions. So is y = 0 or z = 0. Note t hat x = y = z = 0 has b een coun ted 3 times. Hence there are exactly 3 · 2 n − 2 solutions to (8) satisfying xy z = 0. • If xy z 6 = 0, then the n um b er of solutions to (8) is equal to 2 n − 1 m ultiple of that to the system of equations x 2 2 k +1 + y 2 2 k +1 + 1 = x 2 k +1 + y 2 k +1 + 1 = x + y + 1 = 0 (9) with xy 6 = 0. By (9) w e hav e x 2 2 k +1 + ( x + 1) 2 2 k +1 + 1 = x 2 k +1 + ( x + 1) 2 k +1 + 1 = 0 which is equiv alen t to x 2 k = x . Hence x ∈ F 2 k ∩ F ∗ 2 n = F ∗ 2 d and y = x + 1. Since y 6 = 0, then x 6 = 1. Therefore (9) has 2 d − 2 solutions with xy 6 = 0. In total, w e get M 3 = 3 · 2 n − 2 + (2 d − 2)(2 n − 1) = 2 n + d + 2 n − 2 d . 3 The W eight Distr ibution of th e Cyclic Co de C In the sequel we will giv e the t he v alue distribution of S ( α, β , γ ) and, at the same time, the w eigh t distribution of binary cyclic co de C . 8 The or em 1. The value distribution of the multi-set { S ( α, β , γ ) | α, β , γ ∈ F q } and the w eight distribution of C ar e shown as fol lowing (Column 1 is the value of S ( α , β , γ ) , Column 2 is the weight of c ( α, β , γ ) and Column 3 is the c orr esp on ding multiplicity). value weight multiplicity 2 ( n + d ) / 2 2 n − 1 − 2 ( n + d − 2) / 2 (2 n − d − 1 +2 ( n − d − 2) / 2 )(2 n − 1)(2 n +2 d − 2 n − 2 n − d +2 2 d ) 2 2 d − 1 − 2 ( n + d ) / 2 2 n − 1 + 2 ( n + d − 2) / 2 (2 n − d − 1 − 2 ( n − d − 2) / 2 )(2 n − 1)(2 n +2 d − 2 n − 2 n − d +2 2 d ) 2 2 d − 1 2 ( n +3 d ) / 2 2 n − 1 − 2 ( n +3 d − 2) / 2 (2 n − 3 d − 1 +2 ( n − 3 d − 2) / 2 )(2 n − d − 1)(2 n − 1) 2 2 d − 1 − 2 ( n +3 d ) / 2 2 n − 1 + 2 ( n +3 d − 2) / 2 (2 n − 3 d − 1 − 2 ( n − 3 d − 2) / 2 )(2 n − d − 1)(2 n − 1) 2 2 d − 1 0 2 n − 1 (2 n − 1)(2 2 n − 2 2 n − d +2 2 n − 4 d +2 n − 2 n − d − 2 n − 3 d +1) 2 n 0 1 Pr o of. Let n i to b e the n um b er of pa irs ( α, β ) ∈ F q \{ (0 , 0 ) such that r α,β = s − i . Define Ξ =  ( α, β , γ ) ∈ F 3 q | S ( α, β , γ ) = 0  and ξ =   Ξ   . Since n/d is o dd, from Lemma 2, Lemma 3 and Lemma 4 w e ha ve n 1 + n 3 = 2 2 n − 1 (10) n 1 + 2 2 d · n 3 = 2 n − d (2 d + 1)(2 n − 1) . (11) These t w o equations yield n 1 = (2 n − 1)(2 n +2 d − 2 n − 2 n − d + 2 2 d ) 2 2 d − 1 , n 3 = (2 n − d − 1)(2 n − 1) 2 2 d − 1 (12) Note that S (0 , 0 , γ )=0 unless γ = 0. F rom Lemma 2 w e get that ξ = 2 n − 1 + (2 n − 2 n − d ) n 1 + (2 n − 2 n − 3 d ) n 3 = (2 n − 1)(2 2 n − 2 2 n − d + 2 2 n − 4 d + 2 n − 2 n − d − 2 n − 3 d + 1) (13) Then the result follow s from (12), (13) by using Lemma 2. 9 W e hereb y could give the p ossible v a lues of correlation function among se- quences in F . F or example, let a α 1 ,β 1 =  T r n 1 ( α 1 π λ (2 2 k +1) + β 1 π λ (2 k +1) + π λ )  q − 2 λ =0 and a α 2 ,β 2 =  T r n 1 ( α 2 π λ (2 2 k +1) + β 2 π λ (2 k +1) + π λ )  q − 2 λ =0 . Then the correlation func- tion of a α 1 ,β 1 and a α 2 ,β 2 b y a shift τ (0 ≤ τ ≤ q − 2) is C ( α 1 ,β 1 ) , ( α 2 ,β 2 ) ( τ ) = q − 2 P λ =0 ( − 1) a α 1 ,β 1 ( λ ) − a α 2 ,β 2 ( λ + τ ) = q − 2 P λ =0 ( − 1) T r n 1 ( α 1 π λ (2 2 k +1) + β 1 π λ (2 k +1) + π λ ) − T r n 1 ( α 2 π ( λ + τ )(2 2 k +1) + β π ( λ + τ )(2 k +1) + π λ + τ ) = S ( α ′ , β ′ , γ ′ ) − 1 where α ′ = α 1 − α 2 π τ (2 2 k +1) , β ′ = β 1 − β 2 π τ (2 k +1) , γ ′ = 1 − π τ . (14) R emark . As a corollary , w e hav e Cor ol lary 1. The n o n -trivial c orr ela tion values of the se quenc es in F is − 1 , ± 2 n + d 2 − 1 and ± 2 n +3 d 2 − 1 . 4 Conclus ion and F urther Stu dy In this pap er w e ha v e studied the exponential sums S ( α, β , γ ) with α , β , γ ∈ F 2 n . After giving the v alue distribution of S ( α, β , γ ), w e determine the w eigh t distributions of the cyclic co des C . F or the case n/d ev en, w e could get the p ossible v alues of S ( α, β , 0) and S ( α , β , γ ). But the first four momen t iden tities P α,β ∈ F q S ( α , β , 0) i and P α,β ∈ F q S ( α , β , γ ) i for 0 ≤ i ≤ 3 is not enough to determine the v alue distribution of S ( α, β , 0 ) and S ( α , β , γ ). Ho w ev er, w e could get the p ossible no n-triv ail w eigh ts of C : 2 n − 1 , 2 n − 1 ± 2 n 2 + d − 1 and 2 n − 1 ± 2 n 2 +2 d − 1 . New mac hinery and techn ique should b e prop osed to attack this problem. 5 Ac kno wledg emen ts The authors will thank the anonymous referees for their helpful commen ts. 10 References [1] E.F. Assm us Jr and H.F. Mattson Jr, “Some 3- error-correcting BCH codes ha v e corering radius 5,” I EEE T r ans. I nform. The ory, v ol. 22, pp. 3 48–349, Ma y 1976. [2] E.R. Berlek amp, “The w eigh t en umerators fo r certain sub co des of the sec- ond order Reed-Muller codes,” In f o . and Contr ol, vol. 17 , pp. 485–500, 1970. [3] P . Charpin, T. Helleseth and V.A.Zinov iev, “The coset distribution of triple-error-correcting binary primitiv e BCH co des,” IEEE T r ans. Inform . The ory, v ol. 52, no. 4, pp. 1727–173 2 , April 2006. [4] P . Charpin, T. Helleseth and V.A.Zinovie v, “On cosets of we ight 4 of binary BCH co des with minimal distance 8 and exp onential sums,” Pr ob. Inf. T r ans., vol. 41, no. 4, pp. 301–320, 2005. [5] P . Charpin, T. Helleseth and V.A.Zino viev, “On cosets of w eigh t 4 of BCH(2 m ,8), m ev en, and exp onen tial sums,” SIAM J. Discr ete Math., v ol. 23, no. 1, pp. 59–78, 2008. [6] P . Charpin and V.A.Zinoviev , “On cosets of w eigh t distribution o f the 3- error-correcting BCH-co des,” S IAM J. Discr ete Math., v ol. 10, no. 1 , pp. 128–145, 1997. [7] H. Dobb ertin, P . F elk e, T. Helleseth, and P . Rosendahl, “Niho t yp e cross- correlation functions via D ic kson p olynomials and Klo osterman sums,” IEEE T r ans. Inform. The ory, v ol. 52, no. 2, pp. 613–627, F eb. 2006. [8] K. F eng and J. Luo, “V alue distribution of exp onen tial sums from p erfect nonlinear functions and their applications,” IEEE T r an s . Inf. The ory, v ol. 53, no. 9, pp. 3035–30 41, Sept. 2007. [9] T. Helleseth, “P airs of m-sequences with a six-v alued crosscorrelation,” Mathematic al Mathematic al Pr op erties of Se quenc es and Other Combin a- torial Structur es, J.-S. No, H.-Y. Song, T. Helleseth, and P . V. Kumar, Eds. pp. 1–6, Boston: Kluw er Academic Publishers, 2003. [10] T. Hellese th, A. K holosha, and G.J. Ness, “Characterization of m-Sequences of lengths 2 2 k − 1 and 2 k − 1 with three-v a lued cross correlation,” I EEE T r ans. Inform. The ory, v ol. 53, no. 6, pp. 2236–224 5 , June 2 0 07. 11 [11] T. Helleseth and P .V. Kumar, “Sequences with lo w correlation,” in Hand- b o ok of C o ding The ory , V. S. Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: North-Holla nd, 1998. [12] L. Hu, X. Zeng, N. Li and W. Jiang, “P erio d-different m - sequence s with a t most a four-v alued cross correlation,” 11th IEEE Singap ore In ternational Conference on Comm. Systems (ICCS), pp. 44 6–450, No v. 2008. [13] A. Johansen and T. Helleseth, “A family of m- sequences with fiv e-v alued cross correlations”, IEEE T r ans. Inf. T h e ory, v ol. 55, no. 2, pp. 880–887 , F eb. 2009. [14] A. Johansen, T. Hellese th, and X. T ang, “The correlation distribution of quaternary sequence s of p erio d 2(2 n − 1),” IEEE T r ans. I nf. The ory, v ol. 54,no. 7, pp.3130–3139 , July 2008 . [15] T. Kasami, “W eigh t distribution o f Bose-Chaudh uri-Ho cquenghem co des,” in Com b. Math. and Its App.. R . C. Bose and T. A. D o wling, Eds., Chap el Hill, NC: Univ erisit y of North Carolina Press, 1969, pp. 33 5 –357. [16] R. Lidl, and H. Niederreiter, Finite F ields, Addison-W esley , Encyclop edia of Mathematics and its Applications, vol. 20, 1983. [17] J. Luo and K. F eng, “On the weigh t distributions of tw o classes of cyclic co des,” I EEE T r ans. I nf. The ory, vol. 54, no. 12, pp. 5332–5 344, Dec. 2 008. [18] J. Luo and K. F eng, “Cyclic co des and seq uences from g eneralized Coulter Matthews function,” I EEE T r ans. Inf. The ory, v ol. 54, no.12, pp.5345– 5353, Dec. 2008. [19] J. Luo, Y. T ang , H. W a ng “Exp onen tial sums, cyclic co des and sequences: the binary Kasami case,” preprin t. [20] F.J. MacW illiams and N.J.A. Sloane, The Theory of Error-correcting Co des, the 3rd prin ting, North Holland, 1977. [21] M.J. Moisio, “The momen ts o f a Klo o sterman sum and the w eight distri- bution of a Zetterb erg-type binary cyclic code, ” I EEE T r ans. Inf. The ory, v ol. 53, no. 2, pp.843–8 47, F eb.2007. 12 [22] G.J. Ness and T. Helleseth, “ Cross correlation o f m-sequences of differen t lengths,” IEEE T r ans. Inf. T he ory, vol. 52, no. 4, pp. 1637–1 648, Apri. 2006. [23] G. J. Ness and T. Helleseth, “A new family of four-v alued cross correlation b et w een m-sequences of different lengths,” IEEE T r ans. Inf. The ory, v ol. 53, no. 11, pp. 4308–4 313, Nov. 2007. [24] Y. Niho, “Multiv alued cross-correlation functions b et w een tw o maximal lin- ear r ecursiv e sequences,” Ph.D dissertation, Univ. So uth.Calif., Los Angles, 1972. [25] P . Rosendahl, “Niho type cross-correlation functions a nd related equa- tions,” Ph.D dissertation, Depart. Mat h., Univ. T urku, F inland, 2004. [26] M.K. Simon, J. Omura, R. Sc holtz, and K . Levitt, “Spread Sp ectrum Com- m unications,” Ro c kville, MD: Computer Science , 1985 , v ol.ICI I I. [27] M. V an Der Vlugt, “Surfaces and the w eight distribution of a family of co des,” IEEE T r ans. Inf. The ory, v ol. 43 , no. 4, pp. 13 5 4–1360, Apri. 1997. [28] J. W o lfmann, “W eigh t distribution of some binary primitiv e cylcic co des,” IEEE T r ans. Inf. The ory, vol. 40, no. 6, pp. 2068–2 0 71, Jun. 2 0 04. [29] N.Y. Y u a nd G . Gong, “A new binary seq uence family with low correlatio n and lar g e size,”, IEEE T r ans. Inf . The ory, v ol. 52, no. 4, pp. 1624–163 6 , April 2006. [30] N.Y. Y u a nd G . Gong, “New binary sequences with optimal auto correlation magnitude,” IEEE T r ans. In f. The ory, v ol. 54, no. 10, pp. 4771–477 9 , Oct. 2008. [31] X. Zeng, J.Q. Liu, and L. Hu, “Generalized Kasami sequences: the larg e set,” IEEE T r ans. Inf. The ory, vol. 53,no. 7, pp.2587–2598 , July 2007. 13