A Class of Three-Weight Cyclic Codes

A Class of Three-Weight Cyclic Codes Zhengchun Zhou Cunsheng Ding Abstract arxiv:30.0569v [cs.it] 4 Feb 03 Cyclic codes are a subclass of linear codes have alications in consumer electronics, data storage systems, communication systems as they have efficient encoding decoding algorithms. In this aer, a class of threeweight cyclic codes over GF whose duals have two zeros is resented, is an odd rime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are otimal. The duals of a subclass of the cyclic codes are also studied roved to be otimal. Index Terms Cyclic codes, weight distribution, quadratic form, shere acking bound. I. INTRODUCTION Throughout this aer, let m k be ositive integers such that s = m/e is odd s 3, e=gcdm,k. Let be an odd rime q= e. Let π be a rimitive element of the finite field GFq s, q s = m. An [n,l,d] linear code over GF is an l-dimensional subsace of GF n with minimum Hamming distance d. Let A i denote the number of codewords with Hamming weight i in a code C of length n. The weight enumerator of C is defined by +A x+a x + +A n x n. The sequence,a,a,,a n is called the weight distribution of the code C. An [n,l] linear code C over the finite field GF is called cyclic if c 0,c,,c n C imlies c n,c 0,c,,c n C. By identifying the vector c 0,c,,c n GF n with c 0 + c x+c x + +c n x n GF[x]/x n, any code C of length n over GF corresonds to a subset of GF[x]/x n. The linear code C is cyclic if only if the corresonding subset in GF[x]/x n is an ideal of the olynomial residue class ring GF[x]/x n. It is well known that every ideal of GF[x]/x n is rincial. Let C =gx, gx is monic has the least degree. Then gx is called the generator olynomial hx = x n /gx is referred to as the arity-check olynomial of C. A cyclic code is called irreducible if its arity-check olynomial is irreducible over GF. Otherwise, it is called reducible. The weight distributions of both irreducible reducible cyclic codes have been interesting subjects of study for many years. For information on the weight distribution of irreducible cyclic codes, the reader is referred to [8], [3], [9], the recent survey [6]. Information on the weight distribution of reducible cyclic codes could be found in [], [7], [4], [5], [5], [6], []. Let h 0 x, h x, h x be the minimal olynomials of π, π, π k +/ over GF, resectively. It is easy to show that h 0 x, h x, h x are olynomials of degree m are airwise Z. Zhou s research was suorted by the Natural Science Foundation of China, Proj. No. 6043. C. Ding s research was suorted by The Hong Kong Research Grants Council, Proj. No. 6008. Z. Zhou is with the School of Mathematics, Southwest Jiaotong University, Chengdu, 6003, China email: zzc@home.swjtu.edu.cn. C. Ding is with the Deartment of Comuter Science Engineering, The Hong Kong University of Science Technology, Clear Water Bay, Kowloon, Hong Kong, China email: cding@ust.hk.

distinct. The cyclic code over GF with length m arity-check olynomial h 0 xh x has been extensively studied is a three-weight code in the following cases. When k is even e=, this three-weight cyclic code is due to Trachtenberg [0]. When k is odd, e=, =3, the cyclic code is related to some lanar functions is roved to have only three nonzero weights by Yuan, Carlet, Ding [3], []. When k e are odd is any odd rime, Luo Feng [4] roved that the code has only three nonzero weights. The objective of this aer is to study the cyclic code over GFq with length m arity-check olynomial h xh x. It will be shown that this cyclic code has only three nonzero weights when k/e is odd, or k is even e is odd. The weight distribution of the roosed cyclic codes will be determined. Some of the cyclic codes with arity-check olynomials h xh x are otimal. The duals of a subclass of the cyclic codes are also otimal. The three-weight cyclic codes dealt with in this aer may have alications in association schemes [] secret sharing schemes [3]. This aer is organized as follows. Section II introduces necessary results on quadratic forms that will be needed later in this aer. Section III defines the class of cyclic codes determines their weight distributions. Section IV studies the duals of a subclass of the cyclic codes. Section V concludes this aer makes some comments on this toic. II. QUADRATIC FORMS OVER FINITE FIELDS In this section, we give a brief introduction to the theory of quadratic forms over finite fields which is needed to calculate the weight distribution of the cyclic codes in the sequel. Quadratic forms have been well studied see the monograh [7] the references therein, have alications in sequence design [0], [], coding theory [7], [4], [5]. Identifying GFq s with the s-dimensional GFq-vector sace GFq s, a function Q from GFq s to GFq can be regarded as an s-variable olynomial on GFq. The former is called a quadratic form over GFq if the latter is a homogeneous olynomial of degree two in the form Qx,x,,x s = a i j x i x j i j s a i j GFq, we use a basis {β,β,,β s } of GFq s over GFq identify x= s i= x iβ i with the vectorx,x,,x s GFq s. The rank of the quadratic form Qx is defined as the codimension of the GFq-vector sace V ={x GFq s Qx+z Qx Qz=0 for all z GFq s }. That is V =q s r r is the rank of Qx. For a quadratic form fx in s variables over GFq, there exists a symmetric matrix A of order s over GFq such that fx=xax, X =x,,x s GFq s X denotes the transose of X. For a symmetric matrix A of order s over GFq, it is known that there is a nonsingular matrix T of order s such that TAT is a diagonal matrix [7]. Under the nonsingular linear substitution X = ZT with Z =z,z,,z s GFq s, we then have fx=ztat Z = r i= d i z i r is the rank of fx d i GFq. Let = d d d r for r = for r = 0. Let η denote the quadratic multilicative character of GFq. Then η is an invariant of A under the conjugate action of M GL s GFq. The following results are useful in the sequel. Lemma.: [7], [5] With the notations as above, we have ζ Tr q/ fx = x GFq s { η q s r/, if q mod 4, η r q s r/, if q 3 mod 4

3 for any quadratic form fx in s variables of rank r over GFq, ζ is a rimitive -th root of unity, Tr q/ x denotes the trace function from GFq to GF. Lemma.: Let fx be a quadratic form in s variables of rank r over GFq. If r is even, then If r e are odd, then y GF x GFq s y GF x GFq s ζ Tr q/y fx =± q s r/. ζ Tr q/y fx = 0. Proof: By a nonsingular linear substitution as in, we have fx= r i= d iz i, d i GFq z,z,,z r GFq r. Note that, for each y GF, y fx ia a quadratic form over GFq with rank r y fx= r i= yd iz i. According to Lemma., we have Thus, when r is even, ζ Tr q/y fx y GF x GFq s y GF x GFq s On the other h, when r e are both odd, = x GFq s ζ Tr q/ fx ζ Tr q/y fx =± q s r/. ζ Tr q/y fx y GF x GFq s = x GFq s = x GFq s = x GFq s = 0 ζ Tr q/ fx ζ Tr q/ fx ζ Tr q/ fx y GF η y r y GF η y y GF η 0 y y GF η y r. η 0 is the quadratic multilicative character of GF in the third identity we used the fact that η 0 x=η x for any x GF since e is odd. III. THE CLASS OF THREE-WEIGHT CYCLIC CODES AND THEIR WEIGHT DISTRIBUTION We follow the notations fixed in Section I. From now on, we always assume that λ is a fixed nonsquare in GFq. Note that s is odd, thus λ is also a nonsquare in GFq s. Let SQ denote the set of all squares in GFq s. Then λx runs through all nonsquares in GFq s as x runs through SQ. The following result is easy to rove is useful in the sequel. Proosition 3.: λ +k / = λ if k/e is even, λ +k / = λ otherwise. By Delsarte s Theorem [4], the code C with the arity-check olynomial h xh x can be exressed as C ={c GFq s }

4 c = Tr q s / a π t q + bπ k +t/ s. t=0 In terms of exonential sums, the weight of the codeword c =c 0,c,,c q s in C is given by WTc = #{0 t q s : c t 0} = q s q s t=0 = q s y GF = q s = q s = q s y GF ζ yc t q s 3/ t=0 y GF x SQ y GF x SQ = m m y GF x GFq s ζ Tr q s /yaπ t +ybπ t k +/ + ζ Tr q s / yaππ t +ybππ t k +/ ζ Tr q s /yax+ybx k +/ + ζ Tr q s / yaπx+ybπx k +/ ζ Tr q s /yax+ybx k +/ + ζ Tr q s / yaλx+ybλx k +/ y GF x GFq s ζ Tr q s /yax +ybx k+ + ζ Tr q s / yaλx +ybλ k +/ x k + ζ Tr q s /yax +ybx k + + ζ Tr q s / yaλx +ybλ k +/ x k + in the fifth identity we used the fact that both πx λx run through all nonsquares in GF q s as x runs through SQ. It then follows from Proosition 3. that when k/e is even, when k/e is odd, S= T= y GF x GFq s WTc = m m S y GF x GFq s ζ Tr q s /yax +ybx k + + ζ Tr q s / yaλx +ybλx k + ; 3 WTc = m m T ζ Tr q s /yax +ybx k + + ζ Tr q s / yaλx ybλx k +. 4 Based on the discussions above, the weight distribution of the code C is comletely determined by the value distribution of S T. To calculate the value distribution of S T, we need a series of lemmas. Before introducing them, we define for each GFq s. Q x=tr q s /qax + bx +k, x GFq s. 5

5 Lemma 3.: [7], [5] For any GFq s \{0,0}, the function Q of 5 is a quadratic form over GFq with rank s,s, or s. Lemma 3.3: Let k be even e be odd, let S be defined by 3. Then for any 0,0, S takes on only the values from the set {0,± m+e/ }. Proof: According to the definition of S, we have S= y GF x GFq s ζ Tr q/yq x + ζ Tr q/yλq x Q x is given by 5. We now rove that at least one of the quadratic forms Q Q has rank s. When b = 0, it is easy to check that both Q Q have rank s for any nonzero a. When b 0, suose on the contrary that both Q Q have rank less than m. Then there are two nonzero elements x,x GFq s such that Note that Q x + z Q x Q z=0, z GFq s 6 Q x + z Q x Q z=0, z GFq s. 7 Q x+z Q x Q z=tr q s /qzax+bx k + b k x k. It then follows from 6 7, resectively, that b k x k + a k x k + bx = 0 b k x k a k x k + bx = 0. Combining these two equations the first one times x k lus the second one times x k leads to uu k + =0 8 u=bx x x k + x k. Note that x k for any x GFq s since k/e is even s is odd. It then follows that u 0 u k + 0. This is a contradiction with 8. Thus at least one of the quadratic forms Q Q has rank s for any 0,0. On the other h, by Lemma., S 0 only if Q or Q has even rank. Thus, S=± m+e/ if Q has rank s Q has rank s or Q has rank s Q has rank s, otherwise S= 0. This comletes the roof. Theorem 3.4: Let k be even e be odd. Then the value distribution of S in 3 is given by m occurring time m+e/ occurring m e + m e/ m times m+e/ occurring m e m e/ m times 0 occurring m m e + m times. Proof: It is clear that S0,0= m. According to Lemma 3.3, we define N ε = #{ GFq s \0,0 S= ε m+e/ }

6 ε=±. Then we have On the other h, it follows from 3 that S= m +N N m+e/ 9 S =4 m +N + N m+e. 0 S= m S = m #S + #S + #S 3 + #S 4 S ={y,y,x,x Γ y x y x = 0, y x k + y x k + S ={y,y,x,x Γ y x + λy x = 0, y x k + λy x k + S 3 ={y,y,x,x Γ λy x y x = 0, λy x k + y x k + S 4 ={y,y,x,x Γ λy x + λy x = 0, λy x k + λy x k + = 0}. Herein, Γ=GF GF GFq s GFq s. It is not hard to rove that Combining Equations 9 4, we get #S = #S 4 = m 3 #S = #S 3 =. 4 N = m e + m e/ m, N = m e m e/ m. Summarizing the discussion above comletes the roof of this theorem. Lemma 3.5: Let k/e be odd T be defined by 4. Then for any GFq s \{0,0}, T takes on only the values from the set {0,± m+e/ }. Proof: According to the definition of T, we have T= y GF x GFq s ζ Tr q/yq x + ζ Tr q/ yλq x Q x is given by 5. By Lemma 3., for any 0,0, the ossible rank of Q x is s, s, or s. Note that, for any y GF, the quadratic forms yq x λyq x have the same rank with Q x. When Q x has even rank s, by Lemma., we have T=± m+e/. When Q x has odd rank s or s, we distinguish between the following two cases to show that T= 0. Case : e is odd. In this case, it follows again from Lemma. that T=0. Case : e

7 is even. In this case, is a square in GFq thus λ is a nonsquare in GFq. It then follows from Lemma. that ζ Tr q/yq x + ζ Tr q/ yλq x x GFq s =η y+η yλ ζ Tr q/q x x GFq s = 0 for any y GF. Thus T=0. This comletes the roof. Theorem 3.6: Let k/e be odd. Then the value distribution of T in 4 is given by m occurring time m+e/ occurring m e + m e/ m times m+e/ occurring m e m e/ m times 0 occurring m m e + m times. Proof: It is clear that T0,0= m. According to Lemma 3.5, we define n ε = #{ GFq s \0,0 T=ε m+e/ } ε=±. Then we have T= m + n n m+e/ 5 T =4 m + 4n + n m+e. 6 On the other h, it follows from 4 that T= m 7 T = m #T + #T + #T 3 + #T 4 8 T ={y,y,x,x Γ y x y x = 0, y x k + y x k + T ={y,y,x,x Γ y x + λy x = 0, y x k + + λy x k + T 3 ={y,y,x,x Γ λy x y x = 0, λy x k + y x k + T 4 ={y,y,x,x Γ λy x + λy x = 0, λy x k + + λy x k + = 0}. Herein, Γ=GF GF GFq s GFq s. It is not hard to show that Combining Equations 5 9, we get #T = #T = #T 3 = #T 4 = m. 9 n = m e + m e/ m, n = m e m e/ m.

8 TABLE I WEIGHT DISTRIBUTION OF C FOR EVEN k AND ODD e Hamming Weight Frequency 0 m m m+e / m e + m e/ m m m m m e + m m m + m+e / m e m e/ m TABLE II WEIGHT DISTRIBUTION OF C FOR ODD k/e Hamming Weight Frequency 0 m m m+e / m e + m e/ m m m m m e + m m m + m+e / m e m e/ m The value distribution of Ta, b follows from the discussion above. Theorem 3.7: Let k be even e be odd. Then the code C is a three-weight -ary cyclic code with arameters [ m,m, m m m+e / ]. Moreover the weight distribution of C is given in Table I. Proof: The length dimension of C follow directly from its definition. The minimal distance weight distribution of C follow from Equation 3 Theorem 3.4. The following are some examles of the codes. Examle 3.8: Let =3 m=3, k=. Then the code C is a [6,6,5] code over GF3 with the weight enumerator +3 5 + 60x 8 + 56x. It has the same arameters with the best known cyclic codes in the Database of best linear codes known maintained by Markus Grassl at htt://www.codetables.de/. It is also otimal since the uer bound is 5. Examle 3.9: Let =3, m=5 k=4. Then the code C is a [4,6,53] code over GF3 with the weight enumerator +780 53 + 9844x 6 + 744x 7. It has the same arameters with the best known cyclic codes in the Database. It is otimal or almost otimal since the uer bound on the minimal distance of any ternary linear code with length 4 dimension 6 is 54. Examle 3.0: Let =5, m=3, k =. Then the code C is a [4,6,90] code over GF5 with the weight enumerator +370 90 + 944x 00 + 480x 0. The best known linear code over GF5 with length 4 dimension 6 has minimal distance 95. Theorem 3.: Let k/e be odd. Then the code C is a three-weight -ary cyclic code with arameters [ m,m, m m m+e / ]. Moreover the weight distribution of C is given in Table II. Proof: The length dimension of C follow directly from its definition. The minimal distance weight distribution of C follow from Equation 4 Theorem 3.6.

9 Examle 3.: Let =3 m=6, k=. Then the code C is a [78,,43] code over GF3 with the weight enumerator +3760 43 + 4747x 486 + 608x 540. Examle 3.3: Let =5 m=3, k=. Then the code C is a [4,6,80] code over GF5 with the weight enumerator +860 80 + 54x 00 + 40x 0. Examle 3.4: Let = 3 m = 9, k = 3. Then the code C is a [968,8,879] code over GF3 with the weight enumerator +7439796 636 + 3730730x 3 + 690838x 3608. IV. THE DUALS OF A SUBCLASS OF THE CYCLIC CODES In this section, we study the duals of a subclass of the cyclic codes resented in this aer rove that they are otimal ternary linear codes. Theorem 4.: Let =3, k be even e be odd. Then the dual C of the cyclic code C in is an otimal ternary code with arameters [3 m,3 m m,4]. Proof: We only need to rove that C has minimal distance 4. Clearly, the minimal distance d of the dual of C cannot be. Let u= k + /. Then gcdu, m = since k is even m is odd. By the definition of C, the code C has a codeword of Hamming weight if only if there exist two elements c,c GF3 two distinct integers 0 t < t m such that { c π t + c π t = 0 c π ut + c π ut 0 = 0. Note that gcdu, m = t t. It follows from the second equation of 0 that c = c t = t + m /. Then the first equation becomes c π t = 0, which is imossible. Thus the code C does not have a codeword of Hamming weight. We now rove that C has no codeword of weight 3. Otherwise, there exist three elements c,c,c 3 in GF3 three distinct integers 0 t < t < t 3 m such that { c π t +c π t +c 3 π t 3 = 0 c π ut +c π ut +c 3 π ut 3 = 0. Due to symmetry it is sufficient to consider the following two cases. Case A, when c = c = c 3 = : In this case, becomes { π t + π t + π t 3 = 0 π ut +π ut +π ut 3 = 0. Let x i = π t i for i=,,3. Then x,x,x 3 GF3 m are airwise distinct. Without loss of generality, we only need to consider the following two subcases. t is even t,t 3 are odd. In this subcase, we have { x x x 3 = 0 x u + xu + 3 xu 3 = 0 which yields Thus x + x 3 u = x u xu 3. x + x 3 u = x u xu 3

0 which leads to x x 3 x k / x k / 3 = 0. 4 It then follows that x 3 /x k =. Note that gcdm,k=, x,x 3 are both nonsquare in GF3 m since t,t 3 are odd. Thus x = x 3. This is a contradiction to the fact that x x 3. t, t, t 3 are even. Similarly, in this subcase, we can arrive at 4 in which x,x 3 are both squares in GF3 m since t,t 3 are even. It then follows from x 3 /x k = that x = x 3. This is again a contradiction. Case B, when c = c = c 3 =. The roof of this case is similar to Case A. We omit the details here. Finally, by the Shere Packing bound, the minimal distance d 4. Hence d = 4. This comletes the roof. This subclass of ternary cyclic codes C are otimal in the sense that the minimum distance is maximal for any ternary linear code with length 3 m dimension 3 m m. Examle 4.: Let = 3 m = 3, k =. Then the code C is an otimal ternary cyclic code with arameters [6,0,4] generator olynomial x 6 + x 5 + x 3 + x+. Examle 4.3: Let = 3 m = 5, k = 4. Then the code C is an otimal ternary cyclic code with arameters [4,0,4] generator olynomial x 0 + x 9 + x 8 + x 7 + x 6 + x +. Examle 4.4: Let = 3 m = 7, k = 6. Then the code C is an otimal ternary cyclic code with arameters [86,4,4] generator olynomial x 4 + x 3 + x + x 0 + x 6 + x 5 +. V. SUMMARY AND CONCLUDING REMARKS In this aer, we resented a class of three-weight cyclic codes determined their weight distributions. Some of the codes are otimal, the duals of a subclass of the cyclic codes are also otimal. While a lot of two-weight codes were discovered see [], [5], [6], [3], [0], [9], only a small number of three-weight codes are known [], [5], [6], [8], [9]. It would be good if more three-weight codes are constructed, in view of their alications in association schemes secret sharing schemes. REFERENCES [] A. R. Calderbank W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., vol. 8,. 97-, 986. [] A. R. Calderbank J. M. Goethals, Three-weight codes association schemes, Philis J. Res., vol. 39,. 43 5, 984. [3] C. Carlet, C. Ding, J. Yuan, Linear codes from highly nonlinear functions their secret sharing schemes, IEEE Trans. Inform. Theory, vol. 5, no. 6,. 089 0, 005. [4] P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inform. Theory, vol. IT-, no. 5,. 575 576, Se. 975. [5] C. Ding, Y. Liu, C. Ma, L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, vol. 57, No.,. 8000 8006, Dec. 0. [6] C. Ding J. Yang, Hamming weights in irreducible cyclic codes, Discrete Mathematics, vol. 33, no. 4,. 434 446, 03. [7] K. Feng J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Al., vol. no.,. 390 409, Ar. 008. [8] T. Feng, On cyclic codes of length r with two zeros whose dual codes have three weights, Des. Codes Crytogr., vol. 6,. 53 58, 0. [9] G. McGuire, On three weights in cyclic codes with two zeros, Finite Fields Al., vol. 0, no.,. 97 04, Jan. 004. [0] W. M. Kantor, Exonential numbers of two-weight codes, difference sets symmetric designs, Discrete Mathematics, vol. 46,. 95 98, 983. [] A. Klaer, Cross-correlations of quadratic form sequences in odd characteristic, Des. Codes Crytogr.,, vol. 3, no. 4,. 89 305, 997. [] Z. Liu X.-W. Wu, On a class of three-weight codes with crytograhic alications, In: Proc. of the 0 IEEE International Symosium on Information Theory,. 55 555, 0. [3] Z. Liu X. Zeng, On a kind of two-weight code, Euroean Journal of Combinatorics, vol. 33, no. 6,. 65 7, Aug. 0. [4] J. Luo K. Feng, Cyclic codes sequences from generalized Coulter-Matthews function, IEEE Trans. Inform. Theory vol. 54, no.,. 5345 5353. Dec. 008. [5] J. Luo K. Feng, On the weight distributions of two classes of cyclic codes, IEEE Trans. Inform. Theory, vol. 54, no.,. 533 5344, Dec. 008.

[6] C. Ma, L. Zeng, Y. Liu, D. Feng, C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory, vol. 57, no.,. 397 40, Jan. 0. [7] R. Lidl H. Niederreiter, Finite Fields, Encycloedia of Mathematics, Vol. 0, Cambridge University Press, Cambridge, 983. [8] F. MacWilliams J. Seery, The weight distributions of some minimal cyclic codes, IEEE Trans. Inform. Theory, vol. 7 no. 6,. 796 806, 98. [9] B. Schmidt C. White, All two-weight irreducible cyclic codes, Finite Fields Al., vol. 8,. 7, 00. [0] H. M. Trachtenberg, On the crosscorrelation functions of maximal linear recurring sequences, Ph.D. dissertation, Univ. South. Calif., Los Angels, 970. [] J. Yuan, C. Carlet, C. Ding, The weight distribution of a class of linear codes from erfect nonlinear functions, IEEE Trans. Inform. Theory, vol. 5, no.,. 7 77, Feb. 006. [] B. Wang, C. Tang, Y. Qi, Y. X. Yang, M. Xu, The weight distributions of cyclic codes ellitic curves, IEEE Trans. Inform. Theory, vol. 58, no.,. 753 759, Dec. 0. [3] M. van der Vlugt, On the weight hierarchy of irreducible cyclic codes, J. Comb. Theory Ser. A, vol. 7, no.,. 59 67, July 995.