Complete weight enumerators of two classes of linear codes

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1 Comlete weght enumerators of two classes of lnear codes Quyan Wang, Fe L, Kelan Dng and Dongda Ln 1 Abstract arxv: v1 [cs.it] 23 Dec 215 Recently, lnear codes wth few weghts have been constructed and extensvely studed. In ths aer, for an odd rme, we determned the comlete weght enumerator of two classes of -ary lnear codes constructed from defnng set. Results show that the codes are at almost seven-weght lnear codes and they may have alcatons n secret sharng schemes. Index Terms Lnear code, comlete weght enumerator, Gauss sum. I. INTRODUCTION Throughout ths aer, let ba an odd rme and let q = e for some ostve nteger e. Let F denote the fnte feld wth elements. An [n,k,d] lnear code C s over F s a k-dmensonal subsace of F n wth mnmum dstance d [19]. Let A be the number of codewords of weght n C of length n. The (Hammng weght enumerator of C s defned by [19] 1+A 1 x+a 2 x 2 + +A n x n. For n, the lst A s called the weght dstrbuton or weght sectrum of C. A code C s sad to be a t-weght code f the number of nonzero A wth 1 n s equal to t. Clearly, the mnmum dstance of C can be derved from the weght dstrbuton of the code C. By error detecton and error correcton algorthms [21], the weght dstrbuton of a code can be aled to comute the error robablty of error detecton and correcton. Thus, weght dstrbuton s a sgnfcant research toc n codng theory and was studed n [2], [4], [7], [8], [9], [1], [14], [3], [31]. Let F =,1,..., 1}. For a codeword c = (c,c 1,...,c n 1 C, the comlete weght enumerator of c s the monomal w(c = w t w t 1 1 w t 1 1 n the varables w, w 1,..., w 1. Here t ( 1 s the number of comonents of c whch equal to. Then the comlete weght enumerator of the lnear code C s the homogeneous olynomal CWE(C = c C w(c of degree n (see [27], [28]. The authors are suorted by by a Natonal Key Basc Research Project of Chna (211CB324, Natonal Natural Scence Foundaton of Chna ( , the Strategc Prorty Research Program of the Chnese Academy of Scences, Grant No. XDA6171 and Foundaton of NSSFC(No.13CTJ6. Q. Wang s wth the State Key Laboratory of Informaton Securty, the Insttute of Informaton Engneerng, The Chnese Academy of Scences, Bejng, Chna. Emal: wangquyan@e.ac.cn F. L s the corresondng author and wth School of Statstcs and Aled Mathematcs, Anhu Unversty of Fnance and Economcs, Bengbu Cty, Anhu Provnce, Chna. Emal: cczxlf@163.com K. Dng s wth the State Key Laboratory of Informaton Securty, the Insttute of Informaton Engneerng, The Chnese Academy of Scences, Bejng, Chna. Emal: dngkelan@e.ac.cn D. Ln s wth the State Key Laboratory of Informaton Securty, the Insttute of Informaton Engneerng, The Chnese Academy of Scences, Bejng, Chna. Emal: ddln@e.ac.cn

2 2 From defnton, the comlete weght enumerators of bnary lnear codes are just the weght enumerators. For nonbnary lnear codes, the (Hammng weght enumerators, whch have been extensvely nvestgated [15], [16], [34], [15], [35], can be obtaned from the comlete weght enumerators. Further more, the comlete weght enumerators are closely related to the deceton of some authentcaton codes constructed from lnear codes [11], and used to comute the Walsh transform of monomal functons over fnte felds [18]. Thus, a great deal of research s devoted to the comutaton of the comlete weght dstrbuton of secfc codes [1], [3], [2], [22], [23], [25]. Let F q be the fnte feld wth q elements and D = d 1,d 2,...,d n } be a nonemty subset of F q. A generc constructon of a lnear code of length n s gven by C D = c x = (Tr(xd 1,Tr(xd 2,...,Tr(xd n : x F q }, (1.1 where Tr denote the trace functon from F q onto F [26]. The set D s called the defnng set of C D. Ths constructon technque s emloyed n lots of researches to get lnear codes wth few weghts. The readers are referred to [17], [29], [35], [12], [24], [29] for more detals. Naturally, a generalzaton of the code C D of (1.1 s defned by [32] C D = (Tr(xd 1 +u,tr(xd 2 +u,...,tr(xd n +u : u F, x F q }. (1.2 The objectve of ths aer s to resent lnear codes over F wth at most seven weghts usng the above two constructon methods. Further more, the comlete weght enumerators of the two roosed lnear codes are also calculated. The codes n ths aer may have alcatons n authentcaton codes [13], secret sharng schemes [33] and consumer electroncs. II. THE MAIN RESULTS In ths secton, we only resent the -ary lnear codes and ntroduce ther arameters. The roofs of ther arameters wll be gven later. For a F, the defnng set s gven by D a = x F q : Tr(x α +1 = a}, (2.1 where α s any natural number. It should be remarked that, for = 2, the weght enumerator of C D of (1.1 have been determned n [17]. In ths aer, for a F the comlete weght enumerators of C Da of (1.1 and C Da of (1.2 have been exlctly resented by usng exonental sums. Lemma 1 ( [5],Lemma 2.6: Let d = gcd(α,e and be odd. Then 2, f e/d s odd, gcd( d +1, e 1 = d +1, f e/d s even. Note that gcd( α +1, e 1 = 2 leads to whch means that x α +1 : x F q } = x2 : x F q } D = x F q : Tr(xα +1 = } = x F q : Tr(x2 = }. By Lemma 1, f e/d s odd, the code C D of (1.1 and the code C D n [16] are the same. Hence, we wll assume e/d s even and e = 2m for a ostve nteger m. The man results of ths aer are gven below.

3 3 TABLE I: The weght dstrbuton of the codes of Theorem 2 Weght w Multlcty A 1 ( 1 e 2 e 1 ( 1 m 1 1 ( 1( e 2 m 1 ( 1( e 1 + m 1 Theorem 2: Let m 2. If m/d 1 mod 2, then the code C D of (1.1 s a [ e 1 ( 1 m 1 1,e] lnear code wth weght dstrbuton n Table I and ts comlete weght enumerator s 1 w e 1 ( 1 m 1 1 +( e 1 ( 1 m 1 1w e 2 ( 1 m 1 1 +( 1( e 1 + m 1 w e =1 w e 2 m 1. =1 w e 2 TABLE II: The weght dstrbuton of the codes of Corollary 3 Weght b Multlcty A 1 ( 1 e 2 e 1 ( 1 m 1 1 ( 1( e 2 m 1 ( 1( e 1 + m 1 e 1 ( 1 m ( 1( e 2 m 1 1 ( 1( e 1 ( 1 m 1 1 ( 1 e 2 ( 2 m 1 1 ( 1 2 ( e 1 + m 1 Corollary 3: Let the symbols and condtons be the same as Theorem 2. The code C D of (1.2 s a [ e 1 ( 1 m 1 1,e+1] lnear code wth weght dstrbuton n Table II and ts comlete weght enumerator s 1 = w e 1 ( 1 m ( e 1 ( 1 m ( 1( e 1 + m 1 = = w e 2 1 w e 2 ( 1 m 1 1 j w e 2 m 1 j. j w e 2 j TABLE III: The weght dstrbuton of the codes of Theorem 4. Weght b Multlcty A 1 ( 1 e 2 +2 m ( 1(e 1 + m 1 ( 1 e 2 e 1 2 ( 1(e 1 + m 1 1 Theorem 4: Let g be a generator of F and a F. If m/d 1 mod 2, then the code C D a (1.1 s a [ e 1 + m 1,e] lnear code wth weght dstrbuton n table Table III and ts comlete weght enumerator

4 4 s w e 1 + m 1 + ( e 1 ( 1 m w e 2 + m 1 +( e 1 + m 1 +( e 1 + m β=1 2 1 β=1 w e 2 ( 1 m 1 w e 2 2g β w e 2 2g β +( w e m 1 =1 =1 w e 2 w,±2g β ( e 2 ( w e 2 2 4g 2β+1 m 1, 2 4g 2β m 1 where ( denotes the Legendre symbol. TABLE IV: The weght dstrbuton of the codes of Corollary 5 Weght b Multlcty A 1 ( 1 e 2 +2 m ( 1( 2(e 1 + m 1 ( 1 e 2 e ( 1( 2(e 1 + m 1 1 e 1 + m 1 1 ( 1 e 2 + m 1 ( 1(2 e 1 ( 2 m 1 1 Corollary 5: Let the symbols and condtons be the same as Theorem 4. The code C Da of (1.2 s a [ e 1 + m 1,e+1] lnear code wth weght dstrbuton n Table IV and ts comlete weght enumerator s 1 = w e 1 + m 1 +( e 1 + m 1 +( e 1 + m 1 + ( e 1 ( 1 m ( w e β=1 = 1 β=1 = m ( w e 2 1 m 1 = w e 2 +2g β w e 2 2g β j w e 2 + m 1 j wj j,±2g β w e 2 j ( e 2 ( w e 2 j 2 4g 2β+1 m 1 j. j 2 4g 2β m 1 Examle 1: Let (,m,α = (3,3,1. If a =, then the code C D has arameters [224,6,144] wth comlete weght enumerator 2 2 w w 62 w w 8 w 72, and the C D has arameters [224,7,143] wth comlete weght enumerator w w 62 wj w 8 wj 72. F 3 F 3 j F 3 j If a = 1, then the code C D1 has arameters [252,6,162] wth comlete weght enumerator 2 w w 9 w w 72 w9 1 w9 2, =1 =1 1

5 5 and the code C D1 has arameters [252,7,162] wth comlete weght enumerator w w 9 wj w 72 wj 9. F 3 F 3 j F 3 j TABLE V: The weght dstrbuton of the codes of Theorem 6. Weght b Multlcty A 1 ( 1 e 2 ( 1 2 m+d 2 e e 2d ( 1 e 2 e 2d 1 ( 1 m d 1 1 ( 1 e 2 ( 1 m+d 1 ( 1( e 2d 1 + m d 1 Theorem 6: Let m > d+1. If m/d mod 2, then the code C D of (1.1 s a [ e 1 ( 1 m+d 1 1, e] lnear code wth weght dstrbuton n Table V and ts comlete weght enumerator s 1 w e 1 ( 1 m+d 1 1 +( 1( e 2d 1 + m d 1 w e 2 1 +( e e 2d w e 2 ( 1 m+d =1 w e 2 ( 1 m+d 2 + ( e 2d 1 ( 1 m d 1 1 w e 2 ( 1 m+d =1 =1 w e 2 m+d 1 w e 2. TABLE VI: The weght dstrbuton of the codes of Corollary 7. Weght b Multlcty A 1 ( 1 e 2 ( 1 2 m+d 2 e e 2d ( 1 e 2 e 2d 1 ( 1 m d 1 1 ( 1 e 2 ( 1 m+d 1 ( 1( e 2d 1 + m d 1 e 1 ( 1 m+d ( 1 e 2 ( 2 m+d 1 1 ( 1 2 ( e 2d 1 + m d 1 ( 1( e 2 ( 1 m+d 2 1 ( 1( e e 2d ( 1( e 2 m+d 1 1 ( 1( e 2d 1 ( 1 m d 1 1 Corollary 7: Let the symbols and condtons be the same as Theorem 6. The code C D of (1.2 s a [ e 1 ( 1 m+d 1 1,e+1] lnear code wth weght dstrbuton n Table VI and ts comlete weght enumerator s 1 = w e 1 ( 1 m+d ( 1( e 2d 1 + m d 1 1 +( e e 2d = = w e 2 1 w e 2 ( 1 m+d ( e 2d 1 ( 1 m d = j j w e 2 m+d 1 j w e 2 ( 1 m+d 2 j w e 2 ( 1 m+d 1 1 j w e 2 j.

6 6 TABLE VII: The weght dstrbuton of the codes of Theorem 8 Weght b Multlcty A 1 ( 1( e 2 + m+d 2 e e 2d ( 1 e (+1e 2d ( 1m d 1 1 ( 1 e 2 +2 m+d ( 1(e 2d 1 + m d 1 Theorem 8: Let a F. If m/d mod 2, then the code C Da of (1.1 s a [ e 1 + m+d 1,e] lnear code wth weght dstrbuton n Table VII and ts comlete weght enumerator s 1 +( e e 2d w e 1 + m+d 1 = w e 2 + m+d 2 + ( e 2d 1 ( 1 m d 1 1 w e 2 + m+d 1 +( e 2d 1 + m d 1 +( e 2d 1 + m d β=1 2 1 β=1 where ( denote the Legendre symbol. 1 =1 w e 2 w e 2 ( 1 m+d 1 w e 2 2g β w e 2 2g β +( w e m+d 1 =1 w,±2g β ( e 2 ( w e 2 2 4g 2β+1 m+d 1, TABLE VIII: The weght dstrbuton of the codes of Corollary 9. Weght b Multlcty A 1 ( 1( e 2 + m+d 2 ( e e 2d ( 1 e (2 +2( e 2d 1 + m d 1 m d 1 ( 1 e 2 +2 m+d ( 1( 2(e 2d 1 + m d 1 e 1 + m+d 1 1 ( 1 e 2 + m+d 1 ( 1(2 e 2d 1 ( 2 m d g 2β m+d 1 Corollary 9: Let the symbols and condtons be the same as Theorem 8. The code C Da of (1.2 s a [ e 1 + m+d 1,e+1] lnear code wth weght dstrbuton n Table VIII and ts comlete weght enumerator s 1 w e 1 + m+d 1 = 1 +( e e 2d = + ( e 2d 1 ( 1 m d ( e 2d 1 + m d 1 +( e 2d 1 + m d 1 1 = 1 2 ( w e 2 1 β=1 = w e 2 + m+d 2 w e 2 + m+d 1 m+d ( w e 2 1 m+d 1 1 β=1 = j w e 2 j w e 2 +2g β w e 2 2g β j wj j,±2g β ( e 2 ( w e 2 j 2 4g 2β+1 m+d 1 j. j 2 4g 2β m+d 1

7 7 Examle 2: Let (,m,α = (3,4,1. If a =, then the corresondng code C D has arameters [224, 8, 1296] wth comlete weght enumerator w w =1 w w =1 w w =1 w 729, and the code C D has arameters [224,9,1295] wth comlete weght enumerator w w 728 wj w 674 wj w 566 F 3 F 3 j F 3 j F 3 j w 729 j. If a = 1, then the code C D1 has arameters [2268,8,1458] wth comlete weght enumerator 2 2 w w w 81 w w 648 w1 81 w2 81, = and the code C D1 has arameters [2268,9,1458] wth comlete weght enumerator w w w 81 wj w 648 wj 81. F 3 F 3 F 3 j F 3 j =1 III. THE PROOFS OF THE MAIN RESULTS Our task n ths secton s to rove results n Secton 2. Frstly, we revew some basc notatons and results of grou characters and resent some lemma whch are needed for the roofs of the man results. A. Prelmnares We start wth the addtve character. A grou homomorhsm χ from F q nto the comlex numbers s called an addtve character of F q. Let b F q, the mang χ b (c = ζ Tr(bc for all c F q, defnes an addtve character of F q, where ζ = e 2π 1. The addtve character χ s called trval and the other characters χ b wth b F q are called nontrval. The character χ 1 s called the canoncal addtve character of F q. And χ b (x = χ 1 (bx for all x F q. By the orthogonal roerty of addtve characters, we have ([26], Theorem 5.4, q, f χ s trval, χ(x =, f χ s nontrval. The multlcatve characters of F q are Characters over F q, whch are gven by ψ j (g k = ζ 2π 1jk/(q 1 for k =,1,...,q 2, j q 2. Here g s a generator of F q ([26]. The multlcatve character ψ (q 1/2 s called the quadratc character of F q, whch s denoted by η. And we assume that η( = n ths aer. We defne the quadratc Gauss sum G = G(η,χ 1 over F q by G(η,χ 1 = x F q and the quadratc Gauss sum G = G(η,χ 1 over F by G(η,χ 1 = x F η(xχ 1 (x, η(xχ 1 (x,

8 8 where η and χ 1 denote the quadratc and canoncal character of F. The exlct values of quadratc Gauss sums are gven as follows. Lemma 1 ([26], Theorem 5.15: Let the symbols be the same as before. Then G(η,χ 1 = ( 1 (m 1 1 ( 1 2 m 4 q, G(η,χ1 = 1 ( Lemma 11 ([16], Lemma7: Let the symbols be the same as before. Then 1 f m 2 s even, then η(y = 1 for each y F ; 2 f m s odd, then η(y = η(y for each y F. Lemma 12 ([26], Theorem 5.33: Let χ be a nontrval addtve character of F q, and let f(x = a 2 x 2 + a 1 x+a F q [x] wth a 2. Then χ(f(x = χ ( a a 2 1 (4a 2 1 η(a 2 G(η,χ. and For a F, b F q, defne For a, c F and b F q, defne B(a,c = y F A(a = y F ζ ytr(x α +1 (3.1 S(a,b = χ ( ax α +1 +bx. (3.2 z F cz Lemma 13 ([5], Theorem 2: Let e/d be even wth e = 2m. Then m, f a q 1 d +1 1, S(a, = m+d, f a q 1 d +1 = 1, m, m+d, χ 1 (yx α+1 +bzx. (3.3 f a q 1 d +1 1, f a q 1 d +1 = 1, Lemma 14 ([6], Theorem 1: Let q be odd and suose f(x = a α X 2α + ax s a ermutaton olynomal over F q. Let x be the unque soluton of the equaton f(x = b α (b. The evaluaton of S(a, b arttons nto the followng two cases: 1 If e/d s odd, then ( 1 e 1 qη( aχ S(a,b = 1 (ax α +1 2 If e/d s even, then e = 2m, a q 1 d +1 ( 1 m d and, f 1 mod 4, ( 1 e 1 3e qη( aχ 1 (ax α +1, f 3 mod 4. S(a,b = ( 1 m d m χ 1 (ax α +1. Lemma 15 ([6], Theorem 2: Let q be odd and suose f(x = a α X 2α +ax s not a ermutaton olynomal over F q. Then for b we have S(a,b = unless the equaton f(x = b α s solvable. If the equaton s solvable, wth some soluton x say, then S(a,b = ( 1 m d m+d χ 1 (ax α +1. Lemma 16: Let the symbols be the same as before. ( 1 1 If a =, then A( = m+d, f m/d mod 2, ( 1 m, f m/d 1 mod 2.

9 9 2 If a, then A(a = Proof: Notce that ζ ytr(xα +1 m+d, f m/d mod 2, m, f m/d 1 mod 2. = S(y,. For y F, we know y q 1 d +1 = 1. Then the results can be obtaned drectly from Lemma 13. Lemma 17: Let the symbols be the same as before. If m/d 1 mod 2, then X 2α +X = b α has a soluton γ n F q and 1 2 for c, we have 3 for a, we have B(a, = B(, = B(,c = m ( 1 2, f Tr(γ α +1 =, m ( 1, f Tr(γ α +1 ; ( 1 m, f Tr(γ α +1 =, m, f Tr(γ α +1 ; ( 1 m, f Tr(γ α +1 =, m+1 η( atr(γ α +1 m, f Tr(γ α +1 ; 4 for ac, we have m, f Tr(γ α +1 =, B(a,c = m, f Tr(γ α +1 = c 2 /(4a, m+1 η(c 2 4aTr(γ α +1 m, otherwse. Proof: If m/d 1 mod 2, by Theorem 4.1 n [7], we get the equaton X 2α +X = has no soluton nf q. ThenX 2α +X s a ermutaton olynomal overf q andx 2α +X = b α has a unque soluton γ n F q. Thus y 1 ( zγ s the unque soluton n F q of y α X 2α + yx = (bz α. By Lemma 14, S(y,bz = m χ 1 y(y 1 zγ α +1. Therefore, B(a,c = ( m cz χ 1 y(y 1 zγ α +1 y F = m y F z F z F cz 2 Tr(γ α +1 ζ z y. If Tr(γ α +1 =, the corresondng results follow from the orthogonal roerty of addtve characters easly. If Tr(γ α +1, we have B(, = m z F 2 Tr(γ α +1 ζ z y = m z F ζ y = ( 1m. B(,c = m z F y F y F 2 Tr(γ α +1 ζ cz ζ z y y F = m z F ζ cz ζ y = m. y F

10 1 By Lemma 12, we get Also by Lemma 12, we have B(a, = m y F = m y F = m y F = m y F B(a,c = m y F z F ζ ay χ 1 (η 2 Tr(γ α +1 ζ z y = m η(atr(γ α +1 G y F ( +1 χ 1 Tr(γα z 2 y z F ( +1 χ 1 Tr(γα z 2 y z F ( +1 Tr(γα y = m η(atr(γ α +1 G 2 m = m+1 η( atr(γ α +1 m. = m y F = m y F cz z F 2 Tr(γ α +1 ζ z y ( +1 χ 1 Tr(γα y z F ( yc 2 χ 1 η 4Tr(γ α +1 = m η ( Tr(γ α +1 G y F + m y F G m ζ ay η( ay m z 2 cz + m y F ( +1 Tr(γα y c 2 4aTr(γ α +1 y G m 4Tr(γ ζ α +1 η(y m. (3.4 If Tr(γ α +1 = c 2 /(4a, by (3.4, we have B(a,c = m. Otherwse, we obtan B(a,c = m η ( 4aTr(γ α +1 c 2 G y F = m η ( ( c 2 4aTr(γ α +1 G 2 m = m+1 η ( c 2 4aTr(γ α +1 m. ζ c 2 4aTr(γ α +1 4Tr(γ α +1 y ( c 2 4aTr(γ α +1 η y m 4Tr(γ α +1 The roof of ths lemma s comleted. Lemma 18: Let the symbols be the same as Lemma 17. Let m/d mod 2. If X 2α + X = b α has no soluton n F q, then B(a,c =. Suose X 2α +X = b α has a soluton γ n F q, we have 1 B(, = m+d ( 1 2, f Tr(γ α +1 =, m+d ( 1, f Tr(γ α +1.

11 11 2 f c, then 3 f a, then B(a, = B(,c = ( 1 m+d, f Tr(γ α +1 =, m+d, f Tr(γ α +1. ( 1 m+d, f Tr(γ α +1 =, m+d+1 η ( atr(γ α +1 m+d, f Tr(γ α +1, 4 f ac, then m+d, f Tr(γ α +1 =, B(a,c = m+d, f Tr(γ α +1 = c 2 /(4a, m+d+1 η ( c 2 4aTr(γ α +1 m+d, otherwse. Proof: If m/d mod 2, by Theorem 4.1 n [7], we know that X 2α + X s not a ermutaton olynomal. Then usng Lemma 15, the roof s smlar to that of Lemma 17. We omt the detals. Lemma 19: Set f(x = X 2α +X and S = b F q : f(x = b α s solvable n F q }. If m/d s even, then S = e 2d. Proof: Note that both e/d and m/d are even. By Theorem 4.1 n [7], t s easy to see that f(x = has 2d solutons n F q. So does the equaton f(x = b α wth b S. For b 1,b 2 F q and b 1 b 2, t s not hard to know that the two equatons f(x = b α 1 and f(x = b α 2 have no common solutons. Furthermore, for each a F q, there must exst b F q such that f(a = b α, snce X α s a ermutaton olynomal over F q. So the desred result can be concluded wth the above dscussons. B. Proofs of theorems n Secton A Let For a F, recall that we set D a = x F q : Tr(xα +1 = a}. n a = Da }, f a =, D a, f a. From the defnton of n a, we know n a = 1 ytr(xα +1 ay y Fζ Therefore, where A(a s defned by (3.1. For a, c F and b F q, defne = e y F ζ ytr(x α +1. n a = e A(a, (3.5 N b (a,c = x F q : Tr(x α +1 = a and Tr(bx = c}. Let wt(c b denote the Hammng weght of the codeword c b (b F q of the code C D a. It s not dffcult to see that wt(c b = n a N b (a,c. (3.6

12 12. From defnton, for b F q, and a, c F we have N b (a,c = 2 ytr(xα +1 ay y Fζ = 2 1+ y F = e y F + 2 y F z F ζ ytr(xα +1 ay z Fζ ztr(bx cz ζ ytr(x α +1 ay 1+ z F + 2 z F ζ Tr(yx α +1 +bzx ay cz ζ ztr(bx cz ζ ztr(bx cz = e A(a+ 2 B(a,c (3.7 where A(a and B(a, c are defned by (3.1 and (3.3, resectvely. Our task n the sequel s to calculate n a, N b (a,c and gve the roofs of the man results. 1 The frst case: a = and m/d 1 mod 2. Let γ be the unque soluton of equaton X 2α +X = b α. By (3.7, Lemmas 16 and 17, we have the followng two lemmas. Lemma 2: Let b F q, then N b (, = e 2 ( 1 m 1, f Tr(γ α +1 =, e 2, f Tr(γ α +1. Lemma 21: For b F q and c F, we have N b (,c = e 2, f Tr(γ α +1 =, e 2 m 1, f Tr(γ α +1. Now t comes to rove Theorem 2. Proof: By (3.6 and Lemma 16, we get n = e 1 ( 1 m 1. Usng Lemma 2, we have wt(c b ( 1 e 2,( 1 e 2 ( 1 m 1 }. Because wt(c b for each b F q, the dmenson of C D s e. Suose b 1 = ( 1 e 2, b 2 = ( 1 e 2 ( 1 m 1. Also by Lemma 2 and the value of n, we know that A b1 = e 1 ( 1 m 1 1, A b2 = ( 1( e 1 m 1. Hence, we get the Table 1. By the above two lemma, t s easy to get the comlete weght enumerator of C D. And we comlete the roof of Theorem 2.

13 13 2 The second case: a, m/d 1 mod 2. By (3.5, (3.7 and Lemmas 16 and 17, t s easy to get the values of n a, N b (a, and N b (a,c. Lemma 22: For a F, f m/d 1 mod 2, then n a = e 1 + m 1. Lemma 23: For b F q and a F, f m/d 1 mod 2, then N b (a, = e 2 + m 1, f Tr(γ α +1 =, e 2 m 1 η( atr(γ α +1, f Tr(γ α +1. Lemma 24: For b F q, a and c F, f m/d 1 mod 2, then e 2, f Tr(γ α +1 =, N b (a,c = e 2, f Tr(γ α +1 = c 2 /(4a, e 2 m 1 η ( c 2 4aTr(γ α +1, otherwse. Now we begn to rove Theorem 4. Proof: In ths case, usng the above three lemma, as the roof of Theorem 2 one can rove Theorem 4 smlarly. The detals are omtted. 3 The thrd case: a = and m/d mod 2. By (3.5, (3.7and Lemmas 16 and 18, we get the followng two lemmas. Lemma 25: Let b F q and m/d mod 2. If X2α +X = b α has no soluton n F q, then N b (, = e 2 ( 1 m+d 2. If X 2α +X = b α has a soluton γ n F q, then N b (, = e 2 ( 1 m+d 1, f Tr(γ α +1 =, e 2, f Tr(γ α +1. Lemma 26: Let b F q, c F and m/d mod 2. If X2α +X = b α has no soluton n F q, then N b (,c = e 2 ( 1 m+d 2. If X 2α +X = b α has a soluton γ n F q, then N b (,c = e 2, f Tr(γ α +1 =, e 2 m+d 1, f Tr(γ α +1. Now we gve the roof of Theorem 6. Proof: By (3.5 and Lemma 16, we get n = e 1 ( 1 m+d 1. Together wth Lemma 25, we have that wt(c b has three nonzero values. Suose b 1 = ( 1 e 2 ( 1 2 m+d 2, b 2 = ( 1 e 2, b 3 = ( 1 e 2 ( 1 m+d 1. By Lemma 19, A b1 = e e 2d. By the frst two Pless Power Moments([19],. 26 the frequency A b of w satsfes the followng equatons: Ab1 +A b2 +A b3 = m 1, b 1 A b1 +b 2 A b2 +b 3 A b3 = e 1 (3.8 ( 1n, where n = e 1 ( 1 m+d 1 1. Solvng the equatons gves the weght dstrbuton of Table 5. By Lemma 26, together the defnton of comlete weght enumerator of codes, we can obtan the comlete weght enumerator f C D. The roof s comleted.

14 14 4 The fourth case: a and m/d mod 2. By (3.5, (3.7, Lemmas 16 and 18, we have the followng three lemmas. Lemma 27: Let a F and m/d mod 2. Then n a = e 1 + m+d 1. Lemma 28: Let b F q, a F and m/d mod 2. If X 2α + X = b α has no soluton n F q, then N b (a, = e 2 + m+d 2. If X 2α +X = b α has a soluton γ n F q, then N b (a, = e 2 + m+d 1, f Tr(γ α +1 =, e 2 m+d 1 η ( atr(γ α +1, f Tr(γ α +1. Lemma 29: Let a, c F and b F q. Let m/d mod 2. If X2α +X = b α has no soluton n F q, then N b (a,c = e 2 + m+d 2. If X 2α +X = b α has a soluton γ n F q, then e 2, f Tr(γ α +1 =, N b (a,c = e 2, f Tr(γ α +1 = c 2 /(4a, e 2 m+d 1 η ( c 2 4aTr(γ α +1, otherwse. Now we begn to rove Theorem 8. Proof: In ths case, by the three lemmas above, we can get the results n Theorem 6. And we omt the detals. By the defnton of C Da, Corollares 3, 5, 7 and 9 follow drectly from Theorems 2, 4, 6 and 8, resectvely. We omt the roofs. IV. CONCLUDING REMARKS In ths aer, the comlete weght enumerators of two famles of lnear code wth few weghts are determned. Let w mn and w max denote the mnmum and maxmum nonzero weght of a lnear code, resectvely. As ntroduced n [33], any lnear code over F can be emloyed to construct secret sharng schemes wth nterestng access structures f w mn w max > 1. It can be verfed that the lnear code n Theorem 2, Corollary 3 and Theorem 4 have the roerty w mn w max > 1 f m 3. REFERENCES [1] S. Bae, C. L, Q. Yue, On the comlete weght enumerator of some reducble cyclc codes. Dscret. Math. 338 ( [2] L. D. Baumert, R. J. McElece, Weghts of rreducble cyclc codes. Inf. Control 2(2 ( [3] I. F. Blake, K. Kth, On the comlete weght enumerator of Reed-Solomon codes. SIAM J. Dscret. Math. 4(2 ( [4] S. -T. Cho, J. -Y. Km, J. -S. No, and H. Chung, Weght dstrbuton of some cyclc codes, n Proc. Int. Sym. Inf. Theory ( [5] R. S. Coulter, Exlct evaluaton of some Wel sums, Acta Arth, 83 ( [6] R. S. Coulter, Further evaluaton of Wel sums, Acta Arth, 3 ( [7] B. Courteau, J. Wolfmann, On trle sum sets and two or three weghts codes, Dscrete Mathematcs 5 ( [8] W. Chu, C. J. Colbourn, P. Dukes, On constant comoston codes. Dscret. Al. Math. 154 ( [9] C. Dng, A class of three-weght and four-weght codes, n: C. Xng, et al. (Eds., Proc. of the Second Internatonal Worksho on Codng Theory and Crytograhy, n: Lecture Notes n Comuter Scence, 5557, Srnger Verlag, 34 42, 29. [1] C. Dng, Lnear codes from some 2-desgns, IEEE Trans. Inf. Theory 61(6 ( [11] C. Dng, T. Helleseth, T. Kløve, X. Wang, A general constructon of authentcaton codes. IEEE Trans. Inf. Theory 53(6, (27. [12] C. Dng, J. Luo, H. Nederreter, Two-weght codes unctured from rreducble cyclc codes, n: Y. L, et al. (Eds., Proceedngs of the Frst Worsho on Codng and Crytograhy, World Scentfc, Sngaore, , 28. [13] C. Dng and X. Wang, A codng theory constructon of new systematc authentcaton codes, Theoretcal Comut. Sc.,33(1 ( [14] C. Dng, J. Yang, Hammng weghts n rreducble cyclc codes, Dscrete Math. 313(4 (

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