On relations between certain exponential sums and multiple ...

On relations between certain exponentialsums and multiple Kloosterman sumsand some applications to coding theoryMarko MoisioAbstract. In this paper we consider some relations between multiple Kloostermansums and certain exponential sums with monomial and binomial arguments,and some applications of these relations to coding theory.1. IntroductionLet F = F q denote the finite field with q elements and E = F qm be an extension ofdegree m>1ofF . In this paper we consider relations between certain exponentialsumsS(f) := ∑ e E (f(x)),x∈Eand (multiple) Kloosterman sumsK m (a) :=∑x 1 ,...,x m ∈F ∗ e F (x 1 + ···+ x m + ax −11 ...x −1m ),where e E (resp. e F ) is the canonical additive character of E (resp. F ), f(X) ∈E[X], a ∈ F ,gcd(q, deg f(X)) = 1, and some applications of these relations tocoding theory.More precisely, we shall construct a class of binary irreducible cyclic codes andtwo classes of binary cyclic codes (by means of certain well known irreducible cyclic1

2codes), and estimate the weights of the words of these codes by using the Weil andthe Deligne bounds, obtained by Weil in [12] and by Deligne in [1]:|S(f)| ≤(deg f(X) − 1)q m/2 , (1)|K m (a)| ≤(m +1)q m/2 . (2)It turns out e.g. that the weight distribution of the binary cyclic code constructedby means of two simplex codes of different length is closely related to the values ofa multiple Kloosterman sum over a field of characteristic 2.2. On relations between certain exponentialsums and multiple Kloosterman sumsWe assume that F , E, e F , e E and K m () are fixed as in the introduction. Wealso denote the canonical additive character of E (resp. F )simplybye if this doesnot cause any confusions.to F .Let Tr E/F and N E/F denote the trace and norm mappings, respectively, from ELet K be any finite field. We denote the multiplicative character group of K by ̂Kand the identity element of that group by χ 0 .Ifψ is an additive character of K andχ ∈ ̂K, we denote the Gauss sum over K associated to these characters by G K (χ, ψ).If e is the canonical additive character of K, wedenoteG K (χ) =G K (χ, e). Bynotation e a (a ∈ K) we mean the character defined by e a (x) =e(ax) for all x ∈ K.We do not define χ(0), and consequently Gauss sums are calculated over K ∗ .Let ψ be a non-trivial additive character of K. The orthogonality relations ofcharacters, see [4, p. 195], implyψ(x) =1|K|−1∑G(χ, ψ)χ(x) ∀x ∈ K ∗ , (2.1)χ∈ ̂KWe shall also need the following two results:Theorem 2.1. Let d ||K| −1, andletH denote the subgroup of order d of ̂K.Then∑x∈K ∗ ψ(ax d )= ∑ χ∈HG(χ, ψ)χ(a) ( = ∑ χ∈HG(χ, ψ)χ(a) )

3for all a ∈ K ∗ .Proof. See [4, p. 217].Lemma. Let γ be a primitive element of K, a ∈ K and d ||K|−1. Then∑e(ax d )=dx∈K ∗|K|−1d∑−1e(aγ di )=d∑e(x).i=0x∈aProof. Obvious.Let d | q − 1andH (resp. H ′ ) be the subgroup of order d of ̂F (resp. Ê). Thesurjectivity of N E/F implies H ′ = {χ ◦ N E/F | χ ∈ H}.Denote N = N E/F . By Theorem 2.1 and the Davenport-Hasse theorem [4, p.197-199], we now have∑e E (ax d )= ∑ G E (χ ◦ N)χ(N(a)) = (−1) ∑ m−1 G F (χ) m χ(N(a)). (2.2)x∈E ∗ χ∈H χ∈HAssume now that d = q − 1 and consequently H = ̂F . Let x 1 ,...,x m−1 ∈ F ∗and denote b =N(a). It follows from (2.1) thate F (bx −11 ...x −1m−1 )= 1q − 1∑G F (χ)χ(x 1 ) ...χ(x m−1 )χ(b).χ∈ ̂FBy multiplying both sides of the preceding equation by e F (x 1 + ···+ x m−1 )andbysumming we obtain∑e F (x 1 + ···+ x m−1 + bx −11 ...x −1m−1 )= 1 ∑G F (χ) m χ(b).q − 1x 1 ,...,x m−1 ∈F ∗Thus we have provedTheorem 2.2.∑χ∈ ̂Fx∈E ∗ e(ax q−1 )=(−1) m−1 (q − 1)K m−1 (N(a)) ∀ a ∈ E ∗ .The surjectivity of N,the Weil bound and the Deligne bound, now imply

4Corollary 2.3.|K m (a)| ≤min{q m+12 − q m+12 − 1, (m +1)q m/2 } ∀ a ∈ F ∗ .q − 1This result is, of course, valid for any non-trivial additive character ψ of F andalso for sums∑ψ(a 1 x 1 + ···+ a m x m−1 + bx −11 ...x −1m−1 ), ∀ a i,b∈ F ∗ .x 1 ,...,x m−1 ∈F ∗Mordell [10] proved that |K m (a)| ≤ q (m+1)/2 ,whenq is prime. Thus our estimategeneralizes and slightly improves this bound.Corollary 2.4. Let a ∈ E ∗ and d | q − 1. Then∑∣ e(ax d ) ∣ ≤ min{(d − 1)q m/2 +1,m( √ q − 1/ √ q)q m/2 }.x∈E ∗Proof. Let γ be a primitive element of E. DenoteT =< γ q−1 > and t =(q − 1)/d.We now have a partition = ⋃ t−1i=0 γdi T . By the Lemma and Theorem 2.2, wehave∑x∈E ∗ e(ax d )=d∑t−1∑i=0 x∈γ di T∑t−1e(ax) =(−1) m−1 d K m−1 (N E/F (aγ di )).i=0The claim follows now from the Deligne bound and the Weil bound.□Next we shall consider certain exponential sums with binomial arguments. Weneed the following result which is a generalization of a result of C.J. Moreno andO. Moreno [11, Theorem 9].We remark that the result can also be proved by using the results of R.J.McEliecein [8], where he used Gauss sums to determine the weight distributions of certainirreducible cyclic codes. We give, howewer, a direct proof for its simplicity andshortness. The proof rely on the 19th century theorem of Stickelberger, consideringthe values of certain Gauss sums [4, p. 202-203].

Assume 2 | m. Then there exists an intermediate field M of E over F satisfying[M : F ]=2.Theorem 2.5. Let a ∈ E ∗ and assume that m is even. If d | q +1,then⎧∑ ⎨ (−1) m/2 q m/2if ind a ≢ k (d),e(ax d )=⎩(−1) m/2−1 (d − 1)q m/2 if ind a ≡ k (d),x∈Ewhere k =0if(1) 2 | q; or2 ∤ q and m ≡ 0(4);or2 ∤ q, m ≡ 2(4)and 2 | (q +1)/d,and k = d/2 if(2) 2 ∤ q, m ≡ 2(4)and 2 ∤ (q +1)/d.5Proof. Let H be the subgroup of order d of ̂M. By (2.2) we have∑e(ax d )=(−1) ∑m/2−1x∈Eχ∈H ∗ G M (χ) m/2 χ(N(a)),where H ∗ := H \{χ 0 } and N := N E/M .Let χ ∈ H ∗ . Since ord(χ) | q + 1, we observe that Stickelbergers theorem isapplicable.Now, if 2 | q or 2 | m/2, then G M (χ) m/2 = q m/2 . To consider the remainingcases, we fix a generator of ̂M, sayλ, and denote t =(q 2 − 1)/d.Now χ = λ tj for some j ∈{1,...,d− 1}. Since ord(χ) =d/ gcd(d, j), we seethat (q +1)/ ord(χ) iseven,if(q +1)/d is even. Consequently, G M (χ) m/2 = q m/2 ,if (q +1)/d is even.Thusinthecase(1)wehave∑d−1∑e(ax d )=(−1) m/2−1 q m/2 λ tj (N(a)).x∈Ej=1Inthecase(2)(q +1)/ ord(χ) is even if and only if j is even. Thus∑d−1∑e(ax d )=(−1) m/2−1 q m/2 (−1) j λ tj (N(a)).x∈Ej=1By observing that, if γ is a primitive element of E then N(γ) is a primitiveelement of M, we easily obtain the result. □

6Proposition 2.6. Assume that m is even. Then⎧∑e(ax qm −1 ⎨ q m − 1 if Tr E/M (a) =0,q+1 )=⎩ − qm − 1x∈E ∗ q +1 K (1 NM/F (Tr E/M (a)) ) if Tr E/M (a) ≠0.Proof. Denote Tr = Tr E/M ,N=N E/M and d =(q m − 1)/(q +1). Ase E (ax d )=e M (Tr(a)N(x) q−1 ), we see that∑e E (ax d )= qm − 1 ∑eq 2 M (Tr(a)x q−1 ).− 1x∈E ∗ x∈M ∗The claim follows now from Theorem 2.2.□Theorem 2.7. Let a, b ∈ E, b ≠0and assume that m is even. Then⎧∑e(ax qm −1 ⎨ − 1 if Tr E/M (a) =0,q+1 +bx) =⎩x∈E(−1) m ∗ 2 −1 (−1) me(±c)q m 2 −1 q m 2 +12 + K 1 (h), if Tr E/M (a) ≠0,q +1where c = ab −(qm −1)/(q+1) , h =N M/F (Tr E/M (c)) and the ” − ” sign holds if andonly if 2 ∤ q and m ≡ 2(4).Proof. Denote t =(q m − 1)/(q +1) andTr=Tr E/M . If Tr(a) =0,thene(ax t )=e M (Tr(a)x t ) = 1 for all x ∈ E, and the claim follows. Assume that Tr(a) ≠0.Since the mapping x ↦→ bx is a permutation of E ∗ , it is enough to consider sumsS := ∑x∈E ∗ e(cx t + x).Let us fix a primitive element of E, sayγ. Denote T =< γ q+1 >. Now we havea partition E ∗ = ⋃ j∈J γj T ,whereJ := {0,...,q}. Denote J ∗ = J \{k}, wherek =(q +1)/2 if2∤ q and m ≡ 2 (4), and otherwise k =0. NowS = ∑ e(cγ tj ) ∑j∈Jx∈γ j Te(x) =e(±c) ∑x∈γ k Te(x)+ ∑j∈J ∗ e(cγ tj ) ∑where the ” − ” sign holds if and only if 2 ∤ q and m ≡ 2(4).Since∑x∈γ j Te(x) = 1 ∑e(γ j x q+1 )q +1x∈E ∗x∈γ j Te(x),

7by the Lemma, it follows from Theorem 2.5 thatS = e(±c)((−1)m/2−1 q m/2+1 − 1)q +1+ (−1)m/2 q m/2 − 1( ∑ e(cγ tj ) − e(±c)).q +1j∈JThe claim follows now easily from the Lemma and Proposition 2.6.□Consider next the sumsS(a, b) := ∑e(ax qm −1q−1+ bx) a, b ∈ E, q > 1.x∈E ∗Assume b ≠ 0. Again it is enough to consider sums S(c, 1), where c = ab −(qm −1)/(q−1) .Denote Tr = Tr E/F and N = N E/F . Assume that Tr(a) ≠0. Letγ be a primitiveelement of E, and denote T =< γ q−1 > and g =N(γ). Now∑q−2S(c, 1) = e(cg i ) ∑i=0x∈γ i TIt follows from the Lemma and Theorem 2.2 thatThus∑x∈γ i Te(x).e(x) =(−1) m−1 K m−1 (g i ).S(c, 1) = (−1) m−1 ∑ q−2i=0 e F (Tr(c)g i )K m−1 (g i )=(−1) m−1∑x 1 ,...,x m ∈F ∗ e F (x 1 + ···+ x m−1 +Tr(c)x m + x −11 ...x −1m−1 x m)= −1+(−1) m−1 ∑x 1 ,...,x m−1 ∈F ∗ e F (x 1 + ···+ x m−1 ) ∑The next two theorems follows now easily.x m ∈Fe F((Tr(c)+x−11 ...x −1m−1 )x m).Theorem 2.8. Let a, b ∈ E, b ≠0.Then∑{ − 1 if a + a q =0,e(ax q+1 + bx) =x∈E ∗ − qe F (−b q+1 (a + a q ) −1 ) − 1 if a + a q ≠0.

8Theorem 2.9. Let a, b ∈ E, b ≠0. Assume that m>2. Then{∑e(ax qm −1 − 1 if TrE/F (a) =0,q−1 +bx) =x∈E (−1) m−1 qK ∗ m−2 (− N E/F (b)Tr E/F (a) −1 ) − 1 if Tr E/F (a) ≠0.The discussion concerning sums S(a, b) is completed by a (trivial)Proposition 2.10. If q>2 then⎧∑e(ax qm −1 ⎨ q m − 1 if Tr E/F (a) =0,q−1 )=⎩ − qm − 1if Trx∈E ∗ E/F (a) ≠0.q − 1Remark. From the result of Theorem 2.8 it is easy to determine the distributionof the even correlations of a set of PN-sequences, so called small Kasami set, bothin binary and non-binary cases. (c.f. [2], [6]).3. Three examplesLet us first consider some basic facts about the trace codes. For a more completedescription see [2].We restrict our considerations to the binary trace codes. Let K be the finitefield with 2 k elements and fix a primitive element of K, sayγ. Let Tr denote thetrace mapping from K to the prime field F 2 . Let P be an additive subgroup ofK[X] andn ∈ Z + . Define a trace code C(P ):={c(f) | f ∈ P }, wherec(f) =(Tr(f(1)), Tr(f(γ)),...,Tr(f(γ n−1 ))) and n ∈ Z + . It is easy to see that wt(c), the(Hamming) weight of a codeword c ∈ C(P ), is equal to 1 2 (n − ∑ n−1i=0 e K(f(γ i ))) [2].The next theorem is also from [2].Theorem 3.1. Set 2 k − 1=nN. The dual of the binary cyclic code B of length nwith zeros γ Ns 1,...,γ Ns uis the code C(P ), whereP = { ∑ ui=1 a ix Ns i| a i ∈ K}.Let q =2 r .ChooseK = F qm and P = {ax q−1 | a ∈ K}. It follows from Theorem3.1 that the dual of the binary cyclic code of length n =(q m −1)/(q −1) with zeroes

γ (q−1)2i , i =1, 2,... is the code C(P ). It is easy to see that ord n (2) = rm. ThusC(P ) is an irreducible cyclic code of dimension rm [5, p. 77-78]. Sincen−1∑i=0e(aγ (q−1)i )= 1q − 1∑x∈K ∗ e(ax q−1 )by the Lemma, it follows from Theorem 2.2 and Corollary 2.4, that|2wt(c) − qm − 1q − 1 |≤min{mq(m−1)/2 , √ qq (m−1)/2 − qm/2 − 1q − 1 }for all c ∈ C(P ) \{(0,...,0)}.In the case m =2,C(P ) is the dual of the Zetterberg code [7, p. 206]. This dualhas been studied at least in [9] and [3].9Let us assume that m is even. Denote d =(q m − 1)/(q +1), and choose K = F qmand P = {ax d + bx | a, b ∈ K}. The dual of the binary cyclic code of length q m − 1with zeroes γ d2i ,γ 2i , i =1, 2,... is the code C(P ).Denote P 1 = {bx | b ∈ K} and P 2 = {ax d | a ∈ K}. The weight of any nonzerocodeword of the code C(P 1 )isequaltoq m−1 . By Proposition 2.6 and theDeligne bound there is no codeword of that weight in the code C(P 2 ). Since C(P 1 )and C(P 2 ) are subgroups of C(P )wehaveC(P )=C(P 1 ) ⊕ C(P 2 ). Obviously|C(P 1 )| = q m , and by Proposition 2.6 |C(P 2 )| = q 2 . Thus |C(P )| = q m+2 . ByProposition 2.6, Theorem 2.7 and the Deligne bound, we know now that there are(1) q m − 1 codewords of weight q m /2 in the code C(P ),(2) q 2 − 1codewordsc whose weights satisfyq m − 1q +1 ≤|2wt(c) − qm +1|≤2 √ q qm − 1q +1 ,(3) (q 2 − 1)q m − q 2 +1codewords c whose weights satisfy|2wt(c) − q m +1|≤q m/2 +2 √ q qm/2 +1q +1 .We remark that C(P 1 )isthesimplexcodeoflengthq m − 1, and C(P 2 )isthecode constructed by pasting together d copies of each codewords of the dual of theZetterberg code of length q +1.

Let us assume that q>2andm>2. Denote d =(q m − 1)/(q − 1), and chooseK = F qm and P = {ax d + bx | a, b ∈ K}. NowC(P ) is the dual of the binary cycliccode of length q m − 1 with zeroes γ d2i ,γ 2i , i =1, 2,....Denote P 1 = {bx | b ∈ K} and P 2 = {ax d | a, b ∈ K}. Again C(P )=C(P 1 ) ⊕C(P 2 ), and |C(P 1 )| = q m . By Proposition 2.10 |C(P 2 )| = q. Thus|C(P )| = q m+1 .By Theorem 2.9, Proposition 2.10 and Corollary 2.3, we now know that there are10(1) q m − 1 codewords of weight q m /2,(2) q − 1 codewords of weight (q m − 1+(q m − 1)/(q − 1))/2.in the code C(P ).For the remaining (q − 1)q m − q + 1 non-zero codewords c ∈ C(P )itholdsthat(3) q − 1 ≤|2wt(c) − q m +1|≤min{q m+12 − q m+12 −qq−1+1, (m +1)q m/2 }.We remark that C(P 1 ) is again the binary simplex code of length q m − 1, andC(P 2 ) is the code constructed by pasting together d copies of each codewords of thebinary simplex code of length q − 1.References1. Deligne P. (1977) Applications de la formule des traces aux sommes trigonometriques.SGA 4 1/2 Springer Lecture Notes in Math 569: 168-232. Springer, NewYork.2. Honkala I. & Tietäväinen A. Codes and Number Theory. In Brualdi R.A., Huffman W. C. & Pless V. (ed) Handbook of Coding Theory. ElsevierScience Publisher, Amsterdam. In Preparation.3. Lachaud G. & Wolfman J. (1990) The weights of the orthogonals of theextended quadratic binary Goppa codes. IEEE Trans. Inform. Theory 36:686-692.4. Lidl R. & Niederreiter H. (1984) Finite Fields. Cambridge Univ. Press,Cambridge.

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