Codes Associated with Special Linear Groups and Power Moments of Multi-dimensional Kloosterman Sums

1 Codes Associated with Special Linear Groups and Po wer Moments of Multi- dimension al Kloosterman Sums Dae San Kim, Member , I EE E Abstract — In this paper , we construct the binary linear codes C ( S L ( n, q )) associated with fi nite sp ecial linear groups S L ( n, q ) , with both n,q powers of t wo. Then, via Pless p ower moment id en- tity and u tilizing our p re vious result on the explicit expression of the Gauss sum f or S L ( n, q ) , we obtain a recursi ve formula f or the power moments of mu lti-dimensional Kloosterman su ms in t erms of t he frequencies of weights i n C ( S L ( n, q )) . In particul ar , when n = 2 , this giv es a recursive f ormula for the power moments of Kloosterman sums. W e illustrate our results with some examples. Index T erms — Kloosterman sum, finite special lin ear gro u p, Pless power moment id entity , weight distribution, Gauss sum. I . I N T RO D U C T I O N Let ψ be a nontrivial add iti ve character o f the finite field F q with q = p r elements ( p a prim e), and let m be a positive integer . Then th e m -dimension al Kloosterman sum K m ( ψ ; a ) ([ 9]) is defined b y K m ( ψ ; a ) = X α 1 , ··· ,α m ∈ F ∗ q ψ ( α 1 + · · · + α m + aα − 1 1 · · · α − 1 m ) ( a ∈ F ∗ q ) . In p articular, if m = 1 , then K 1 ( ψ ; a ) is si m ply d enoted by K ( ψ ; a ) , and is ca lled the Kloo sterman sum. The Kloosterma n sum was in troduce d in 1926 [7] to g i ve an estimate for th e Fourier coefficients of mo dular for ms. For each non negativ e integer h , b y M K m ( ψ ) h we will denote the h -th mo ment of the m -dimension al Kloo sterman sum K m ( ψ ; a ) . Namely , it is giv en b y M K m ( ψ ) h = X a ∈ F ∗ q K m ( ψ ; a ) h If ψ = λ is the cano nical add iti ve ch aracter o f F q , th en M K m ( λ ) h will be simply denoted by M K h m . If further m = 1 , for b revity M K h 1 will be indicated b y M K h . The po wer moments of Kloosterman sums can b e used, fo r example, to giv e an estimate fo r the Kloosterm an sums and h av e also been studied to solve a variety of problem s in cod ing theo ry over finite field s of character istic two. If q = p is an odd prim e, f or h ≤ 4 , M K h was ev aluated by Sali ´ e [15]. For details about these, th e r eader is ref erred to Section IV . This work was supported by grant No. R01-2008-000 -11176-0 from the Basic Research Program of the Kore a Science and Engineering Foundat ion. The author is with the Department of Mathematics, Sogang Univ ersity , Seoul 121-742, Kore a(e-mail: dskim@sogang.ac.kr). From n ow on, let us assume that q = 2 r . Carlitz [1] ev aluated M K h , for h ≤ 4 , while Moisio com puted M K 6 in [14]. Recen tly , Moisio was a ble to find explicit expr essions of M K h , for h ≤ 10 (cf. [11]). This was done, via Pless po wer moment identity , by connecting mo ments of Kloosterman sums an d the f requen cies of weig hts in the b inary Zetterberg code of len gth q + 1 , wh ich were known by the work of Sch oof and Vlu gt in [16]. In th is pap er , we adop t M oisio’ s idea to show th e following theorem giving a recursive fo rmula fo r the power mom ents of mu lti-dimension al Kloosterm an sums. T o do that, we construct the b inary linear cod e C ( S L ( n, q )) associated with the sp ecial linear grou p S L ( n, q ) , and express those power moments in terms of th e fre quencies of weights in th e code. Here, in addition to the a ssumption q = 2 r , we restrict n to be n = 2 s . Then, than ks to our previous result on the explicit expression of “Gauss sum” for the spec ial linear group [6], we c an express the weigh t of each codeword in the dual C ⊥ ( S L ( n, q )) of C ( S L ( n, q )) , in term s of ( n − 1) -dimen sional Kloo sterman sum s. Th en our fo rmula follows imm ediately from th e Pless power momen t id entity . Theor em 1: Let n = 2 s , q = 2 r . Then , for all po siti ve integers h , we have the following recursi ve formula for the moments of multi-dime nsional Klo osterman sums M K h n − 1 : q ( n 2 ) h M K h n − 1 = h − 1 X i =0 ( − 1) h + i +1  h i  N h − i q ( n 2 ) i M K i n − 1 + q min { N ,h } X i =0 ( − 1) h + i C i h X t = i t ! S ( h, t )2 h − t  N − i N − t  . (1) Here N = q ( n 2 ) Q n j =2 ( q j − 1) is th e ord er of S L ( n, q ) , and S ( h, t ) indicates the Stirling nu mber of the secon d kind given by S ( h, t ) = 1 t ! t X j =0 ( − 1) t − j  t j  j h . (2) In add ition, { C i } N i =0 denotes the weight distribution o f the code C = C ( S L ( n, q )) , which is given by C i = X Y β ∈ F q  n β ν β  (0 ≤ i ≤ N ) , 2 where the sum r uns over all the sets of n onnegative integers { ν β } β ∈ F q satisfying P β ∈ F q ν β = i and P β ∈ F q ν β β = 0 (an ide ntity in F q ), and n β = |{ g ∈ S L ( n, q ) | tr ( g ) = β }| = q ( n 2 ) − 1 { n Y j =2 ( q j − 1) + 1 + q θ ( β ) } , with θ ( β ) = ( K n − 2 ( λ ; β − 1 ) , β 6 = 0 , 0 , β = 0 . Here we unde rstand that K 0 ( λ ; β − 1 ) = λ ( β − 1 ) . In addition, fr om now on we agr ee that  b a  = 0 , if b < a . Cor ollary 2: Let q = 2 r . Then , for all positiv e integers h , we have the following recursive formu la for the mom ents of Kloosterman sum s M K h : q h M K h = h − 1 X i =0 ( − 1) h + i +1  h i  N h − i q i M K i + q min { N , h } X i =0 ( − 1) h + i C i h X t = i t ! S ( h, t )2 h − t  N − i N − t  . (3) Here N = q ( q 2 − 1) is the order of S L (2 , q ) , S ( h, t ) indicates the Stirling number of the second kind as in (2), and { C i } N i =0 denotes the weight d istribution of the cod e C = C ( S L (2 , q )) , wh ich is giv en b y C i = X  q 2 ν 0  Y tr ( β − 1 )=0  q 2 + q ν β  Y tr ( β − 1 )=1  q 2 − q ν β  (0 ≤ i ≤ N ) , where the sum r uns over all the sets of n onnegative integers { ν β } β ∈ F q satisfying P β ∈ F q ν β = i and P β ∈ F q ν β β = 0 , and the first and second produ ct run respectively o ver the elements β ∈ F ∗ q , with tr ( β − 1 ) = 0 and tr ( β − 1 ) = 1 . I I . P R E L I M I N A R I E S The following no tations will b e u sed throu ghout th is paper except in Section I V , where q is allowed to be an y prim e powers. n = 2 s ( s ∈ Z > 0 ) , q = 2 r ( r ∈ Z > 0 ) , S L ( n, q ) = th e spe cial linea r grou p, N = q ( n 2 ) Q n j =2 ( q j − 1) th e orde r of S L ( n, q ) , T r ( g ) = the m atrix trace fo r g ∈ S L ( n, q ) , tr ( x ) = x + x 2 + · · · + x 2 r − 1 the trace func tion F q − → F 2 , λ ( x ) = ( − 1) tr ( x ) the canonical add iti ve char acter of F q . Let g 1 , g 2 , · · · , g N be a fixed ordering of the elements in S L ( n, q ) . Let C = C ( S L ( n, q )) be the binar y linear code of length N , defined by : C = C ( S L ( n, q )) = { u ∈ F N 2 | u · v = 0 } , (4) where v = ( T r ( g 1 ) , T r ( g 2 ) , · · · , T r ( g N )) ∈ F N q . (5) Theor em 3 (Delsarte, [10]): Let B be a lin ear code over F q . T hen ( B | F 2 ) ⊥ = tr ( B ⊥ ) . From Delsarte’ s theorem, the next result follows immediately . Pr oposition 4: The dual C ⊥ = C ⊥ ( S L ( n, q )) of C = C ( S L ( n, q )) is given by C ⊥ = { c ( a ) = ( tr ( aT r ( g 1 )) , tr ( aT r ( g 2 )) , · · · , tr ( aT r ( g N ))) | a ∈ F q } . The next Prop osition is stated in T heorem 6.1 o f [6]. Bu t we sligh tly mod ified the exp ression there. Pr oposition 5: Let n β = |{ g ∈ S L ( n, q ) | T r ( g ) = β }| , for eac h β ∈ F q . T hen n β = q ( n 2 ) − 1 { n Y j =2 ( q j − 1) − ( q − 1) n − 1 + q δ ( n − 1 , q ; β ) } , where δ ( n − 1 , q ; β ) = |{ ( α 1 , · · · ,α n − 1 ) ∈ ( F ∗ q ) n − 1 | α 1 + · · · + α n − 1 + α − 1 1 · · · α − 1 n − 1 = β }| . The fo llowing coro llary is im mediate fr om Proposition 5 . Cor ollary 6: The map T r : S L ( n, q ) − → F q giv en by g 7− → T r ( g ) is surjectiv e. Pr oposition 7: The map F q − → C ⊥ ( S L ( n, q )) g i ven by a 7− → c ( a ) is an F 2 -linear isom orphism. Pr oof: It is F 2 -linear and s u rjectiv e. Let a b e in the k ern el of the map. Then tr ( aT rg ) = 0 , f or all g ∈ S L ( n, q ) . In view of Corollary 6, tr ( aα ) = 0 , fo r a ll α ∈ F q . As tr : F q − → F 2 is surjective, a = 0 . The next theorem is ab out the Gauss sum f or S L ( n, q ) , and is one of the main results of the paper [6]. Theor em 8: Let ψ b e any nontrivial additi ve character of F q . T hen X g ∈ S L ( n,q ) ψ ( T r ( g )) = q ( n 2 ) K n − 1 ( ψ ; 1) . For the fo llowing lemma, observe that ( n, q − 1 ) = 1 . 3 Lemma 9: The map a 7− → a n : F ∗ q − → F ∗ q is a bijection. For the proo f of the next p roposition and th e following, we borrowed an idea from the pr oof of Theorem 6 .1 in [13]. Pr oposition 10 : For a ∈ F ∗ q , the Hammin g weight of the codeword c ( a ) = ( tr ( aT r ( g 1 )) , tr ( aT r ( g 2 )) , · · · , tr ( aT r ( g N ))) is given by(cf . Pro position 4): w ( c ( a )) = 1 2 ( N − q ( n 2 ) K n − 1 ( λ ; a )) . (6) Pr oof: w ( c ( a )) = 1 2 N X i =1 (1 − ( − 1) tr ( aT r ( g i )) ) = N 2 − 1 2 X g ∈ S L ( n,q ) λ ( aT r ( g )) = N 2 − 1 2 q ( n 2 ) K n − 1 ( ψ ; 1) (Theor em 8, with ψ ( x ) = λ ( ax ) ) = N 2 − 1 2 q ( n 2 ) X x 1 , ··· ,x n − 1 ∈ F ∗ q λ ( ax 1 + · · · ax n − 1 + ax − 1 1 · · · x − 1 n − 1 ) = N 2 − 1 2 q ( n 2 ) X x 1 , ··· ,x n − 1 ∈ F ∗ q λ ( x 1 + · · · + x n − 1 + a n x − 1 1 · · · x − 1 n − 1 ) = N 2 − 1 2 q ( n 2 ) X x 1 , ··· ,x n − 1 ∈ F ∗ q λ ( x n 1 + · · · + x n n − 1 + a n x − n 1 · · · x − n n − 1 ) (7) (by L emma 9) = N 2 − 1 2 q ( n 2 ) X x 1 , ··· ,x n − 1 ∈ F ∗ q λ (( x 1 + · · · + x n − 1 + ax − 1 1 · · · x − 1 n − 1 ) n ) (8) = N 2 − 1 2 q ( n 2 ) X x 1 , ··· ,x n − 1 ∈ F ∗ q λ ( x 1 + · · · + x n − 1 + ax − 1 1 · · · x − 1 n − 1 ) (9) ([9], Th eorem 2.2 3(v)) = N 2 − 1 2 q ( n 2 ) K n − 1 ( λ ; a ) . W e ar e r eady to d etermine δ ( n − 1 , q ; β ) , which ap pears in Proposition 5. Pr oposition 11 : For each β ∈ F q , let δ ( n − 1 , q ; β ) = |{ ( α 1 , · · · ,α n − 1 ) ∈ ( F ∗ q ) n − 1 | α 1 + · · · + α n − 1 + α − 1 1 · · · α − 1 n − 1 = β }| . Then δ ( n − 1 , q ; 0 ) = q − 1 { ( q − 1) n − 1 + 1 } , and, for β ∈ F ∗ q , δ ( n − 1 , q ; β ) = K n − 2 ( λ ; β − 1 ) + q − 1 { ( q − 1) n − 1 + 1 } , where K 0 ( λ ; β − 1 ) = λ ( β − 1 ) by convention. Pr oof: q δ ( n − 1 , q ; β ) = X α 1 , ··· ,α n − 1 ∈ F ∗ q X α ∈ F q λ ( α ( α 1 + · · · + α n − 1 + α − 1 1 · · · α − 1 n − 1 − β )) = X α ∈ F q λ ( − αβ ) X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( αα 1 + · · · + αα n − 1 + αα − 1 1 · · · α − 1 n − 1 ) = X α ∈ F ∗ q λ ( − αβ ) X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( α 1 + · · · + α n − 1 + α n α − 1 1 · · · α − 1 n − 1 ) + ( q − 1) n − 1 = X α ∈ F ∗ q λ ( − αβ ) X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( α 1 + · · · + α n − 1 + αα − 1 1 · · · α − 1 n − 1 ) + ( q − 1) n − 1 (following the steps in (7)-(9)) = X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( α 1 + · · · + α n − 1 ) × X α ∈ F ∗ q λ ( α ( α − 1 1 · · · α − 1 n − 1 − β )) + ( q − 1) n − 1 = X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( α 1 + · · · + α n − 1 ) × X α ∈ F q λ ( α ( α − 1 1 · · · α − 1 n − 1 − β )) − X α 1 , ··· ,α n − 1 ∈ F ∗ q λ ( α 1 + · · · + α n − 1 ) + ( q − 1) n − 1 = q X λ ( α 1 + · · · + α n − 1 ) + 1 + ( q − 1) n − 1 (10) The sum in (10) runs over all α 1 , · · · , α n − 1 ∈ F ∗ q satisfying α − 1 1 · · · α − 1 n − 1 = β , so th at it is g iv en by      0 , if β = 0 , K n − 2 ( λ ; β − 1 ) , if β 6 = 0 , and n > 2 , λ ( β − 1 ) , if β 6 = 0 , and n = 2 . So we get the desired result. 4 Combining Pr opositions 5 and 11, we get the fo llowing corollary . Cor ollary 12: Let n β = |{ g ∈ S L ( n, q ) | T r ( g ) = β }| , for eac h β ∈ F q . T hen n β = q ( n 2 ) − 1 { n Y j =2 ( q j − 1) + 1 + q θ ( β ) } , (11) where θ ( β ) = ( K n − 2 ( λ ; β − 1 ) , β 6 = 0 , 0 , β = 0 , with th e convention that K 0 ( λ ; β − 1 ) = λ ( β − 1 ) . I I I . P RO O F O F M A I N R E S U LT S In this section, we will derive the rec ursiv e form ula (1) for the power moments of m ulti-dimen sional Klo osterman sums which is expressed in terms of the fr equencies C i of weigh ts in th e code C = C ( S L ( n, q )) . Theor em 13 (P less power moment identity , [10]): Let B be an q -ary [ n , k ] cod e, and let B i (resp. B ⊥ i ) denote the number of codewords of weight i in B (resp. in B ⊥ ). T hen, for h = 0 , 1 , 2 , · · · , n X i =0 i h B i = min { n,h } X i =0 ( − 1) i B ⊥ i h X t = i t ! S ( h, t ) q k − t × ( q − 1) t − i  n − i n − t  , (12) where S ( h, t ) is the Stirling number o f the second kin d defined in (2). Theor em 14 ([8 ]): Let q = 2 r , with r ≥ 2 . Then the r ange R of K ( λ ; a ) , as a varies over F ∗ q , is g i ven by R = { t ∈ Z | | t | < 2 √ q , t ≡ − 1( mod 4) } . In ad dition, each value t ∈ R is attained exactly H ( t 2 − q ) times, wh ere H ( d ) is the Kron ecker c lass num ber of d . Theor em 15 ([2 ]): For the ca nonical additive ch aracter λ of F q , and a ∈ F ∗ q , K 2 ( λ ; a ) = K ( λ ; a ) 2 − q . (13) Let u = ( u 1 , · · · , u N ) ∈ F N 2 , with ν β 1’ s in the coord inate places where T r ( g j ) = β , for ea ch β ∈ F q . Then we see fro m the defin ition of the cod e C = C ( S L ( n, q )) (cf. ( 4),(5)) that u is a codeword with weight i if and only if P β ∈ F q ν β = i an d P β ∈ F q ν β β = 0 (an id entity in F q ). As th ere are Q β ∈ F q  n β ν β  many such co dew o rds with weigh t i , we obtain the following theorem. Theor em 16: Let { C i } N i =0 be the weig ht distribution of th e code C = C ( S L ( n, q )) . Then, fo r 0 ≤ i ≤ N , C i = X Y β ∈ F q  n β ν β  , (14) where n β is as in (1 1), an d th e sum runs over all the sets of nonnegative integers { ν β } β ∈ F q satisfying X β ∈ F q ν β = i and X β ∈ F q ν β β = 0( an identity in F q ) . (15) Cor ollary 17: Let { C i } N i =0 be the weight distribution of the code C = C ( S L ( n, q )) . T hen, for 0 ≤ i ≤ N , C i = C N − i . Pr oof: Under th e rep lacements ν β → n β − ν β , for all β ∈ F q , the first sum in (15) is chan ged to N − i , while the second one in (15) and th e summands in (14) are left unchan ged. Here the second su m in (15) is left unchan ged, since P β ∈ F q n β β = 0 , as one can see by using the explicit expression of n β in (11). Cor ollary 18: Let { C i } N i =0 be th e weight distribution of th e code C = C ( S L (2 , q )) . T hen, f or 0 ≤ i ≤ N , C i = X  q 2 ν 0  Y tr ( β − 1 )=0  q 2 + q ν β  Y tr ( β − 1 )=1  q 2 − q ν β  , where the su m runs over all the sets of nonnegative in tegers { ν β } β ∈ F q satisfying P β ∈ F q ν β = i and P β ∈ F q ν β β = 0 , and th e first and seco nd p roduct ru n respecti vely over the elements β ∈ F ∗ q , with tr ( β − 1 ) = 0 and tr ( β − 1 ) = 1 . Pr oof: For n = 2 , we see from (11) tha t n β is given b y n β =      q 2 , if β = 0 , q 2 + q , if tr ( β − 1 ) = 0 , q 2 − q , if tr ( β − 1 ) = 1 . Cor ollary 19: Assume that r ≥ 2 , and that { C i } N i =0 is the weight distribution of th e code C = C ( S L (4 , q )) . Then , fo r 0 ≤ i ≤ N , C i = X  m 0 ν 0  Y | t | < 2 √ q t ≡− 1(4) Y K ( λ ; β − 1 )= t  m t ν β  , (16) where the su m runs over all the sets of nonnegative in tegers { ν β } β ∈ F q satisfying P β ∈ F q ν β = i and P β ∈ F q ν β β = 0 , m 0 = n 0 = q 5 { 4 Y j =2 ( q j − 1) + 1 } , and m t = q 6 { q 2 ( q 2 − 1)( q 4 − q − 1) + t 2 } , for all integers t satisfying | t | < 2 √ q and t ≡ − 1(4) . 5 Pr oof: No te he re that, for n = 4 , and β ∈ F ∗ q , n β = q 5 { 4 Y j =2 ( q j − 1) + 1 + q K 2 ( λ ; β − 1 ) } = q 5 { 4 Y j =2 ( q j − 1 ) + 1 + q ( K ( λ ; β − 1 ) 2 − q ) } ( cf . ( 13 )) (17 ) = q 6 { q 2 ( q 2 − 1)( q 4 − q − 1) + K ( λ ; β − 1 ) 2 } . Now , in voking Theo rem 14, we obtain the result. W e are now re ady to prove Th eorem 1 , which is the main result of th is paper . T o do th at, we app ly Pless power moment identity in (12), with B = C ⊥ ( S L ( n, q )) . The n, in view of Proposition 7 and u tilizing (6 ), the lef t hand side of (12) is giv en by X a ∈ F ∗ q w ( c ( a )) h = 1 2 h X a ∈ F ∗ q ( N − q ( n 2 ) K n − 1 ( λ ; a )) h = 1 2 h h X i =0 ( − 1) i  h i  N h − i q ( n 2 ) i M K i n − 1 = 1 2 h ( − 1) h q ( n 2 ) h M K h n − 1 + 1 2 h h − 1 X i =0 ( − 1) i  h i  N h − i q ( n 2 ) i M K i n − 1 . On the other hand, n oting that dim C ⊥ ( S L ( n, q )) = r (cf . Proposition 7), the right ha nd side of (12) is given by q min { N ,h } X i =0 ( − 1) i C i h X t = i t ! S ( h, t )2 − t  N − i N − t  . Here th e frequen cies C i of codewords with we ight i in C = C ( S L ( n, q )) are given by (14). Now , Cor ollary 2 follows fr om Theorem 1 and Corollary 18, an d Coro llary 20 f rom Th eorem 1 and Coro llary 19. Cor ollary 20: For all p ositi ve in tegers h , we have the following recursive for mula for th e mo ments of the 3- dimensiona l Kloo sterman sums M K h 3 , q 6 h M K h 3 = h − 1 X i =0 ( − 1) h + i +1  h i  N h − i q 6 i M K i 3 + q min { N ,h } X i =0 ( − 1) h + i C i h X t = i t ! S ( h, t )2 h − t  N − i N − t  . Here N = q 6 Q 4 j =2 ( q j − 1 ) is the ord er of S L (4 , q ) , { C i } N i =0 denotes t h e weig ht distrib ution of the code C = C ( S L (4 , q )) giv en b y ( 16), and S ( h, t ) indicates the Stirling number o f the second k ind as in (2). I V . R E M A R K S Here we will b riefly r evie w th e previous results on power moments of Kloosterman sums M K h , and make some comments on our result in (3). For any q = p r ( p a prime) , M K h = q 2 q − 1 A h − ( q − 1) h − 1 + 2( − 1) h − 1 , (18) where A h = |{ ( α 1 , · · · , α h ) ∈ ( F ∗ q ) h | h X j =1 α j = 0 = h X j =1 α − 1 j }| . For h ∈ Z ≥ 0 , d efine M h as: M h = |{ ( α 1 , · · · , α h ) ∈ ( F ∗ q ) h | h X j =1 α j = 1 = h X j =1 α − 1 j }| , for h > 0 , and M 0 = 0 . Then, as one can see, ( q − 1 ) M h − 1 = A h , for any positive integer h . So (1 8) can b e r e wr itten as M K h = q 2 M h − 1 − ( q − 1) h − 1 + 2( − 1) h − 1 ( h ≥ 1) . (1 9) Sali ´ e obtained th is form o f expression for M K h already in [15], f or any odd prime q . Iwaniec [5 ] showed the expre ssion (18) fo r any p rime q . Howe ver , the proof given there works for a ny p rime power q , withou t any r estriction. Also, this is a special case o f Theorem 1 in [3], a s mentioned in Rem ark 2 there. Let q = p be a ny p rime. Th en M K 1 = 1 , M K 2 = p 2 − p − 1 , M K 3 = ( − 3 p ) p 2 + 2 p + 1 ( with th e un derstandin g ( − 3 2 ) = − 1 , ( − 3 3 ) = 0) , M K 4 = ( 2 p 3 − 3 p 2 − 3 p − 1 , if p ≥ 3 , 1 , if p = 2 . Sali ´ e o btained these results in [15] by d etermining M 1 , M 2 , M 3 , a nd Iwaniec got these o nes in [5] by computin g A 2 , A 3 , A 4 . Except [1] for 1 ≤ h ≤ 4 and [14] fo r h = 6 , no t much progr ess had b een ma de un til Mo isio succe eded in ev alua ting M K h , fo r the othe r values of h with h ≤ 1 0 over the finite fields of char acteristic two (Similar resu lts exist a lso over th e finite fields o f charac teristic thr ee [4],[12]). His results ar e as follows: 6 M K 1 =1 , M K 2 = q 2 − q − 1 , M K 3 =( − 1) r q 2 + 2 q + 1 , M K 4 =2 q 3 − 2 q 2 − 3 q − 1 , M K 5 =( u 1 + ( − 1) r 4) q 3 + 5 q 2 + 4 q + 1 , M K 6 =5 q 4 − (5 + ( − 1) r ) q 3 − 9 q 2 − 5 q − 1 , M K 7 =( u 2 + 6 u 1 + ( − 1) r 14 + 1) q 4 + 14 q 3 + 14 q 2 + 6 q + 1 , M K 8 =14 q 5 − (15 + ( − 1) r 7) q 4 − 28 q 3 − 20 q 2 − 7 q − 1 , M K 9 =( u 3 + 8 u 2 + 27 u 1 + 8 + ( − 1) r 48) q 5 + 42 q 4 + 48 q 3 + 27 q 2 + 8 q + 1 , M K 10 =42 q 6 − (51 + ( − 1) r 35) q 5 − 90 q 4 − 75 q 3 − 35 q 2 − 9 q − 1 − u 4 . (20) Here u 1 , u 2 , u 3 , u 4 are the follo win g n umbers which are depend ent up on the extension degree r of F q over F 2 : u 1 =((1 + √ − 15) / 4) r + ((1 − √ − 15) / 4) r , u 2 =(( − 5 + √ − 39) / 8) r + (( − 5 − √ − 39) / 8) r , u 3 =(( − 3 + √ 505 + q − 510 − 6 √ 505) / 32) r + (( − 3 + √ 505 − q − 510 − 6 √ 505) / 32) r + (( − 3 − √ 505 + q − 510 + 6 √ 505) / 32) r + (( − 3 − √ 505 − q − 510 + 6 √ 505) / 32) r , u 4 =( − 12 + 4 √ − 119) r + ( − 12 − 4 √ − 119) r . As we men tioned earlier, these we re obtained, via Pless power moment identity , b y expressing power mom ents o f Kloosterman sums in terms of th e freq uencies of weights in the binary Zetterb erg cod e of length q + 1 . In fact, Moisio used the freque ncies B i in the Z etterberg code for i ≤ 12 , which wer e av ailable in T ab le 6.2 of [16]. Even though it w as a brea kthroug h, it h ad a fe w dr aw- backs. Firstly , the way it is proved is too ind irect, since the frequen cies are expressed in ter ms of th e Eich ler Selbe rg trace formu las for the Hecke operators actin g on certain spaces of cusp fo rms f or Γ 1 (4) . Secondly , the power mome nts of Kloosterman sums ar e ob tained only fo r h ≤ 10 and no t for any highe r order mom ents. On the o ther ha nd, our fo rmula in (3) allows one, at least in principle, to compu te mo ments of all or ders for any given q . Moreover , it gives a recursive formu la not only for power mom ents of Kloosterma n sums but also for th ose of multi-d imensional Kloosterman sums(cf. (1)). Nev er theless, obviously it is go od to have explicit formulas like the ones p resented in (20) . In the next sectio n, we will give some n umerical examples dem onstrating that o ur formu la in (3) is q uite useful for evaluating power m oments of Kloo sterman sums for each given q . V . E X A M P L E S In this section, for small values of i , we com pute, by u sing Corollary 2 and MA GMA, th e freq uencies C i of weights in C ( S L (2 , 2 3 )) and C ( S L (2 , 2 4 )) , and the power moments M K i of Kloosterm an sum s over F 2 3 and F 2 4 . In particular , our r esults confirm those of Moisio’ s gi ven in (20), when q = 2 3 and q = 2 4 . T ABLE I The weight distribu tion of C ( S L (2 , 2 3 )) w frequenc y w frequenc y 0 1 11 1495424065262442956416 1 64 12 61437005346735099526740 2 15844 13 2325154356197975713774208 3 2650560 14 81546484920999191101202360 4 332067914 15 2663851840752718923500482944 5 33207770816 16 81413971883002952517354367429 6 2 761774095732 17 2337059898759141068388769445824 7 196480443747136 18 63230453927539041393172170525052 8 12206347634256355 19 1617368453093893435845237341156928 9 672705382226871680 20 39221184987526914436447793737809822 10 33298916433035363704 21 903954930188715550538753640492641088 T ABLE II The power moments of Kloosterman sums over F 2 3 i M K i i M K i i M K i 0 7 10 9942775 20 9537789199383 1 1 1 11 -48296687 21 -476805777143519 2 55 12 245734951 22 2384279934194455 3 -47 13 -1215920159 23 -11920646525541647 4 871 14 6117864535 24 59605492064000071 5 -2399 15 -30474531407 25 -298020682011124799 6 17815 16 152717030791 26 1490123744982250615 7 -71567 17 -762552032639 27 -74505577201 31373167 8 410311 18 3815859527095 28 372529716149965055 11 9 -1894079 19 -19069999543727 29 -186264309031963608479 T ABLE III The weight distribu tion of C ( S L (2 , 2 4 )) w fr equency w frequenc y 0 1 6 398943240589827320 1 256 7 232184965775802188544 2 520072 8 11821117069839 4115200330 3 706962176 9 5348398 7453818691622983424 4 720560061732 10 21773331292449548118228026776 5 587401078798592 11 8056132578206330016084726166784 T ABLE IV The power moments of Kloosterman sums over F 2 4 i M K i i M K i i M K i 0 15 4 7631 8 13118351 1 1 5 22081 9 72973441 2 239 6 300719 10 604249199 3 289 7 1343329 11 3760049569 7 A CKNO WLED GMENT I would like to thank Mr . 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