COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS JIYOU LI AND DAQING WAN Abstact I this pape, we obtai a explicit fomula fo the umbe of zeo-sum -elemet subsets i ay fiite abelia goup 1 Itoductio Let A be a abelia goup Let D A be a fiite subset of elemets Fo a positive itege 1 ad a elemet b A, let N D (, b) deote the umbe of -elemet subsets S D such that a S a b The decisio vesio of the subset sum poblem ove D is to detemie if thee is a o-empty subset S D such that a S a b, that is, if N D(, b) > 0 fo some 1 This poblem atually aises fom a umbe of impotat applicatios i codig theoy ad cyptogaphy It is a well ow NP-complete poblem, eve i the case whe A is cyclic (fiite o ifiite), o the additive goup of a fiite field F q o the goup E(F q ) of F q -atioal poits o a elliptic cuve E defied ove F q The case that A Z is the basis of the apsac cyptosystem The case A F q is elated to the deep hole poblem of exteded Reed-Solomo codes, see [2] The case A E(F q ) is elated to the miimal distace of exteded elliptic codes, see [1] The mai combiatoial difficulty fo the subset sum poblem comes fom the geat flexibility i choosig the subset D which is i geeal eithe too small o fa fom ay algebaic stuctue Cosequetly, i ode to say somethig sigificat about N D (, b), it is ecessay to put some estictios o the subset D Fom algoithmic poit of view, the idea of dyamic algoithm [3] ca be used to show that N D (, b) ca be computed i polyomial time if the set D is sufficietly lage i the sese that D A c fo some positive costat c Fom mathematical poit of view, ideally, we would lie to have a explicit fomula o a asymptotic fomula This is appaetly too much to hope fo i geeal, eve i the case that D A c fo some positive costat c Howeve, we expect the existece of a asymptotic fomula fo the umbe N D (, b) fo cetai o-tivial values of if D is close to a lage subset with cetai algebaic stuctue Fo example, a old esult of Ramaatha (1945) gives a explicit fomula fo N D (, b) whe D A is a fiite cyclic goup, obtaied usig Ramauja s tigoometic sums Moe ecetly, the authos [5] obtaied a explicit fomula fo N D (, b) i the case whe D A is the additive goup of a fiite field F q (which is a elemetay abelia p-goup) usig etiely diffeet agumets I this pape, we peset a geeal ew appoach which gives a explicit fomula whe D A is ay fiite abelia goup I paticula, this geealizes ad uifies pevious fomulas i this diectio Ou mai esult is the followig theoem Theoem 11 Suppose we ae give the isomophism A Z 1 Z 2 Z s with A 1 s Give b A, suppose (b 1, b 2,, b s ) is the image of b i 1

2 JIYOU LI AND DAQING WAN the isomophism Let N(, b) be the umbe of -subsets of A whose elemets sum to b The N(, b) 1 ( 1) + / Φ(, b), (,) whee Φ(, b) d,( i,d) b i µ(/d) s ( i, d) ad µ is the usual Möbius fuctio defied ove the iteges Rema The aveage size of the umbe N(, b) is ( ) / Thus, the total iput ad output size fo the poblem of computig N(, b) is oughly s log The above fomula gives a algoithm fo computig N(, b) i time which is polyomial i s log This is a detemiistic polyomial time algoithm Whe A is cyclic oe checs that i the above fomula Φ(, b) has a simple fom Φ(, b) d (b,) µ(/d)d ad thus we get the followig fomula, which was fist foud by Ramaatha [9] usig the popeties of the Ramauja s tigoometical sum Some elated esults ca be foud i [7, 10] Coollay 12 Give b Z Let N(, b) be the umbe of -subsets of Z whose elemets sum to b The we have N(, b) 1 ( 1) + / C (b), (,) whee C (b) d (,b) µ(/d)d is Ramauja s tigoometical sum, which ca be also defied as C (m) e 2πim/ I paticula, N(, 0) 1 (,) whee φ is the Eule totiet fuctio,(,)1 ( 1) + / φ(), Coollay 13 Let N(b) be the umbe of subsets of A sum to b Fo coveiece we egad the empty set as a subset sum to 0 The N(b) 1 Φ(, b)2 /, odd I paticula, whe A is cyclic, we have N(b) 1, odd C (b)2 / Futhemoe, if b 0 ad is odd the we get a classical fomula [10] N(0) 1 φ()2 / Rema The aveage size of N(b) is 2 /, ad thus the iput ad output size fo computig N(b) is oughly O(s) The fomula i the above coollay gives a algoithm which computes the umbe N(b) i time that is polyomial i s This is a detemiistic polyomial time algoithm

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 3 Aothe example is to tae A to be a additive subgoup of a fiite field F q of chaacteistic p (o ay fiite dimesioal vecto space ove F p ) I this case, we obtai Coollay 14 Let F q be the fiite field of q elemets with chaacteistic p Let A be ay additive subgoup of F q ad A Fo ay b A, let N(, b) be the umbe of -subsets of A whose elemets sum to b Defie v(b) 1 if b 0, ad v(b) 1 if b 0 If p, the N(, b) 1 If p, the N(, b) 1 + ( 1) + v(b) p /p /p This geealizes the fomula i [5] which wos whe A F q The pape is ogaized as follows We fist peset a sieve fomula via the Möbius Ivesio Fomula i Sectio 2 ad the pove Coollay 12 i Sectio 3 The poof of Theoem 11, Coollay 13 ad Coollay 14 ae give i Sectio 4 2 A distict coodiate sievig fomula The statig poit of ou appoach is the ew sievig fomula discoveed i [6], which sigificatly impoves the classical iclusio-exclusio sieve i some iteestig cases I this sectio, we will give a efomulatio ad a ew poof of this fomula via the M öbius ivesio o a suitable patially odeed set We ecall some basic otatios fo ou poblem Let D be a fiite set, ad let D D D D be the Catesia poduct of copies of D Let X be a subset of D I may situatios we ae iteested i coutig the umbe of elemets i the set X {(x 1, x 2,, x ) X x i x j, i j} (21) Let S be the symmetic goup o {1, 2,, } Each pemutatio τ S factoizes uiquely (up to the ode of the factos) as a poduct of disjoit cycles ad each fixed poit is viewed as a tivial cycle of legth 1 Fo simplicity of the otatio, we usually omit the 1-cycles Two pemutatios i S ae cojugate if ad oly if they have the same type of cycle stuctue (up to the ode) Let C be the set of cojugacy classes of S ad ote that C p(), the patitio fuctio Fo a give τ S, let l(τ) be the umbe of cycles of τ icludig the tivial cycles The sig(τ) ( 1) l(τ) Fo a give pemutatio τ (i 1 i 2 i a1 )(j 1 j 2 j a2 ) (l 1 l 2 l as ) with 1 a i, 1 i s, defie X τ { (x 1,, x ) X, x i1 x ia1,, x l1 x las } (22) Each elemet of X τ is said to be of type τ Thus X τ is the set of all elemets i X of type τ A patially odeed set, also ow as a poset, is a set P with a biay elatio such that: fo all a, b ad c i P, we have that: a a (eflexivity); if a b ad b a the a b (atisymmety); if a b ad b c the a c (tasitivity)

4 JIYOU LI AND DAQING WAN We use the coveiet otatio x < y to mea x y ad x y We also use y x to deote x y The icidece algeba I(P, R)[10] of a patially odeed set P ove a commutative ig R with idetity is the algeba of fuctios f : P P R such that: f(x, y) 0 uless x y; (f + g)(x, y) f(x, y) + g(x, y) (additio); (fg)(x, y) x z y f(x, z)g(z, y) (covolutio) Oe checs that I(P, R) is a associative R-algeba with idetity δ(x, y), which is defied by δ(x, y) 1 if x y ad δ(x, y) 0 othewise The zeta fuctio ζ is defied by ζ(x, y) 1 fo all x y i P ad ζ(x, y) 0 othewise It is easy to chec that this fuctio is ivetible Its ivese is called the Möbius fuctio of P ad is deoted by µ We ca defie µ iductively Namely, µζ δ is equivalet to µ(x, x) 1, fo all x P ; µ(x, y) x z<y µ(x, z) fo all x < y i P The followig ivesio lemma is oe of the most impotat tools i combiatoics We omit the poof sice it ca be foud i may combiatoial boos Lemma 21 (Möbius Ivesio Fomula) Let (P, ) be a fiite patially odeed set Let f, g : P C The g(x) x y f(y), fo all x P if ad oly if f(x) x y µ(x, y)g(y), fo all x P whee µ(x, y) is the Möbius fuctio as defied above Let [] be the set {1, 2,, } Let Π be the set of set patitios of [] Defie a biay elatio o Π as follows: τ δ if evey bloc of τ is cotaied i a bloc of δ Fo istace, {1, 2}{3, 4}{5, 6} {1, 2, 3, 4}{5, 6} ad {1, 3}{2}{4}{5}{6} {1, 2, 3}{4}{5, 6} Oe checs that Π is ideed a patially odeed set To compute the values of the Möbius fuctio µ i Π is a vey otivial ad impotat esult i the theoy of eumeative combiatoics We cite it diectly without poof Fo details please efe to [10] Lemma 22 Deote 1 to be the smallest elemet i Π Fo ay τ Π, let l be the umbe of blocs i τ ad let 1, 2,, l be the cadiality of each bloc of τ, the we have l µ(1, τ) ( 1) i 1 ( i 1)! Now we will pove ou ew fomula via this ivesio fomula Theoem 23 Let X, X τ be defied as i (21) ad (22) The we have X τ S sig(τ) X τ (23)

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 5 Futhemoe, X τ Π l ( 1) i 1 ( i 1)! X τ, whee ( 1, 2,, l ) i the summatio meas the coespodig bloc sizes of τ Poof We ote that fo a set patitio τ Π we ca defie X τ similaly o eve moe atually Fo ay τ Π, defie Xτ to be the set of vectos x X τ such that thee does ot exist δ Π satisfyig τ < δ ad x X δ Recall that τ < δ if τ δ ad τ δ It is easy to chec that X δ δ τ X τ, ad thus by the Möbius Ivesio Fomula give i Lemma 21 we have X δ δ τ µ(δ, τ) X τ I paticula, let δ 1 {1}{2} {}, the X 1 is just X, which was defied as above to be the set of distict coodiate vectos i X Thus we have X 1 τ µ(1, τ) X τ τ Π µ(1, τ) X τ τ Π :( 1, 2,, l ) τ S sig(τ) X τ l ( 1) i 1 ( i 1)! X τ The last equality comes fom a elemetay coutig o the umbe of pemutatios fo a give set patitio of [] Rema: We ema that covesely we ca pove Lemma 22 by ou fomula (23) i a vey simple way Now the symmetic goup S acts o D by pemutig coodiates That is, fo give τ S ad x (x 1, x 2,, x ) D, we have τ x (x τ(1), x τ(2),, x τ() ) Befoe statig a useful coollay, we fist give a defiitio Defiitio 24 Let G be a subgoup of S A subset X D is said to be G-symmetic if fo ay x X ad ay g G, g x X I paticula, a S - symmetic X is simply called symmetic Futhemoe, if X satisfies the stogly symmetic coditio, that is, fo ay τ ad τ i S, oe has X τ X τ povided l(τ) l(τ ), the we call X a stogly symmetic set The sigless Stilig umbe of the fist id c(, i) is defied to be the umbe of pemutatios i S with exactly i cycles Note that c(, 0) 0 It ca also be

6 JIYOU LI AND DAQING WAN defied by the followig classic equality [10]: ( 1) i c(, i)q i (q), (24) i0 whee (x) x(x 1) (x + 1) fo Z + {1, 2, 3, } ad (x) 0 1 It is immediate to get the followig simple fomula i the symmetic case Coollay 25 Let C be the set of cojugacy classes of S If X is symmetic, the X τ C ( 1) l(τ) C(τ) X τ, (25) whee C(τ) is the umbe of pemutatios cojugate to τ Futhemoe, if X is stogly symmetic, the we have X ( 1) i c(, i) X i, (26) whee X i is defied as X τi fo some τ i S with l(τ i ) i ad c(, i) is the sigless Stilig umbe of the fist id 3 Subset sums o cyclic goups I this pape, we idetify a elemet b Z with its least oegative itege epesetative Lemma 31 Let 1, 2,, l, b be elemets i Z ad ( 1, 2,, l, ) d Let M be the umbe of solutios of the followig coguece equatio ove Z 1 x 1 + 2 x 2 + + l x l b mod The M > 0 if ad oly if d b Moeove, if d b, the we have M d l 1 I paticula, if ( 1, 2,, l, ) 1, the M l 1 Poof As ideals i Z, we have 1 Z + + l Z + Z ( 1,, l, )Z dz Thus, M > 0 if ad oly if d b Assume ow that d b The, M is the umbe of solutios of the liea equatio 1 d x 1 + + l d x l b d mod d i the ig Z, which is ( d )l 1 d l d l 1 This is because each solutio i Z d to exactly d l solutios i Z lifts Lemma 32 Let d Let TP d(j) be the umbe of pemutatios i S of j cycles with the legth of its each cycle divisible by d The we have ( 1) j TP d (j) j ( 1) + /d d! (31) /d j1

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 7 Poof A pemutatio τ S is said to be of type (c 1, c 2,, c ) if τ has exactly c i cycles of legth i Let N(c 1, c 2,, c ) be the umbe of pemutatios i S which is of type (c 1, c 2,, c ) We have the followig coutig fomula N(c 1, c 2,, c ) Defie the geeatig fuctio Fom (32), we have C (t 1, t 2,, t ) C (t 1, t 2,, t ) ici ici! 1 c1 c 1!2 c2 c 2! c c! (32) N(c 1, c 2,, c )t c1 1 tc2 2 tc (33)! c 1!c 2! c! (t 1 1 )c1 ( t 2 2 )c2 ( t )c Thus we obtai the followig expoetial geeatig fuctio C (t 1, t 2,, t ) u! 2 t2 eut1+u 2 +u 3 t3 3 + 0 Now let P d(t) TP d(j)tj By (33) we have P d(t) C (0, 0,, t, 0, ), whee t appeas at the idex divisible by d Fo give geeatig fuctio f(x), we deote [x i ]f(x) to be the coefficiet of x i i the fomal powe seies expasio of f(x) The we have [ ] u P d ( ) (t) e t u d d + u2d 2d +! [ ] u e t d log (1 ud )! [ ] u 1! (1 u d ) t/d [ ] u ( j + t/d 1! j j 0 /d + t/d 1! /d ) (u d ) j It is diect to chec that ( 1) P d ( ) ( 1) + /d d!, /d by the followig equality fo all iteges 1 ( 1) Thus (31) follows fom ( 1) j TP d (j) j ( 1) P d ( ) j1 ad the poof is complete

8 JIYOU LI AND DAQING WAN Lemma 33 Let d,,, b be oegative iteges If the we have (, d)µ(/d) 0 d,(,d) b Poof Sice, thee is a pime p such that od p s > 0 ad od p t < s By the defiitio of the Möbius fuctio, if od p (/d) > 1 the µ(/d) 0 Thus we have (, d)µ(/d) (, d)µ(/d) + (, d)µ(/d) d,(,d) b d,(,d) b odpds d,(,dp) b odpds 1 d,(,d) b odpds 1 (, dp)µ(/dp) + (, d)µ(/d) + d,(,d) b odpds 1 d,(,d) b odpds 1 d,(,d) b odpds 1 (, d)µ(/d) (, d)µ(/d) 0 Fist we pove ou mai esult i the cyclic case Theoem 34 Give b Z ad 1 Let N(, b) be the umbe of -subsets of Z whose elemets sum to b The we have N(, b) 1 ( 1) + / µ(/d)d I paticula, (,) N(, 0) 1 (,) whee φ is the Eule totiet fuctio d (,b) ( 1) + / φ(), Poof Let X be the umbe of solutios of the equatio x 1 + x 2 + + x b i Z Sice X is symmetic, by applyig Coollay 25 we have! N(, b) X τ C ( 1) l(τ) C(τ) X τ, (34) whee X τ is defied as i (22) Sice x 1 + x 2 + + x b is liea, by Lemma 31 it is easy to chec that whe (, ) 1 we always have X τ l(τ) 1, whee l(τ) is the umbe of cycles of τ Thus, whe (, ) 1, X is stogly symmetic ad we coclude N(, b) 1! ( 1) i c(, i) X i 1! 1 1! ( 1) i c(, i) i 1 ( 1) i c(, i) i 1 1! () 1 Now, we coside the geeal case Fo each d we deote TP d to be the cojugacy classes i C whose each cycle legth is divisible by d Let CP d to be the

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 9 cojugacy classes i C such that the geatest commo diviso of the cycle legth equals d Sice TP d d CP By the Möbius Ivesio Fomula we have CP d d µ(/d)tp d µ(/d)tp, whee µ is the usual Möbius fuctio defied ove the iteges Note that fo give τ CP d, if (, d) b the X τ 0 ad othewise by Lemma 31 we have X τ (, d) l(τ) 1, whee (, d) is the geatest commo diviso of ad d Thus by (34) we have! N(, b) ( 1) l(τ) C(τ) X τ τ C ( 1) l(τ) C(τ) X τ d τ CP d d,(,d) b τ CP d ( 1) l(τ) C(τ)(, d) l(τ) 1 Fo give d, deote by TP d(j) ad CP d(j), the umbe of pemutatios i TP d ad CP d espectively with j cycles If d ad (, d) b, the τ CP d ( 1) l(τ) C(τ)(, d) l(τ) 1 ( 1) j CP d (j)(, d) j 1 j1 (, d) (, d) (, d) j1( 1) j µ(/d)tp (j) j d µ(/d) ( 1) j TP (j) j d d j1 µ(/d)( 1) + /! The last equality comes fom Lemma 32 Thus we have N(, b) 1 d) d,(,d) b(, µ(/d)( 1) + / d 1 ( 1) + / (, d)µ(/d) d,(,d) b By Lemma 33, if, the d,(,d) b (, d)µ(/d) 0 Thus by a substitutio d of (, d) we have N(, b) 1 ( 1) + / µ(/d)d (,) d (,b)

10 JIYOU LI AND DAQING WAN We otice that the last summatio is exactly the famous Ramauja s tigoometical sum C (b), which ca be also defied as C (m) e 2πim/, ad satisfies the equality [9] Thus we may wite the above fomula as,(,)1 φ() C (m) µ(/(, m)) φ(/(, m)) N(, b) 1 I paticula, whe b 0 we have N(, 0) 1 (,) (,) ( 1) + ( 1) + / C (b) / φ() Now let us coside the subset sum poblem ove Z Coollay 35 Let N(b) be the umbe of subsets of Z sum to b Fo coveiece we egad the empty set as a subset sum to 0 The we have N(b) 1 C (b)2 /, odd Poof Oe checs that the fomula fo N(, b) holds whe 0, ie, coespodig to the empty set Thus we have 1 N(b) N(, b) ( 1) + / C (b) 0 0 (,) 1 C (b) 0, ( 1) + / Note that the last summatio vaishes if is eve ad othewise we have / ( 1) + / / 2 / l Thus 0, N(b) 1, odd l0 C (b)2 / 4 Subset sums o fiite abelia goups Now we tu to pove Theoem 11 The method is vey simila to the poof of the cyclic case

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 11 Poof Let A be a abelia goup of ode By the stuctue theoy of fiite abelia goups we may suppose that A Z 1 Z 2 Z s with 1 2 s ad 1 2 s Give b A ad suppose b (b 1, b 2,, b s ) Let N(, b) be the umbe of -subsets of A whose elemets sum to b Let X be the set of solutios of the equatio x 1 + x 2 + + x b i A Sice X is symmetic, by applyig Coollay 25 we have! N(, b) ( 1) l(τ) C(τ) X τ, (41) τ C whee X τ is defied as i (22) Now we tu to computig X τ Fo give τ S, suppose τ factos ito l cycles with legths a 1, a 2,, a l espectively The X τ equals the umbe of solutios of the liea equatio a 1 x 1 + a 2 x 2 + + a l x l b ove A Fo ay 1 i l we wite x i (x i1, x i2,, x is ) ad b (b 1, b 2,, b s ) whee x ij, b i Z i 1 j s Thus fo computig X τ it suffices to coside the system of equatios ove Z 1, Z 2,, Z s : a 1 x 1i + a 2 x 2i + + a l x li b 1 Z i, fo i 1,, s Note that we have l a i Whe (, ) 1, the (a 1,, a l, ) 1 ad by Lemma 31 we always have X τ ( i ) l(τ) 1 l(τ) 1, whee l(τ) is the umbe of cycles of τ Thus, i the case (, ) 1, we coclude N(, b) 1 ( 1) i c(, i) X i 1 ( 1) i c(, i) i 1!! 1 1! ( 1) i c(, i) i 1 1! () 1 Fo each d we deote TP d to be the cojugacy classes i C whose each cycle legth is divisible by d Let CP d to be the cojugacy classes i C such that the geatest commo diviso of the cycle legth equals d Let (, d) be the geatest commo diviso of ad d Oe checs that CP d d µ(/d)tp, whee µ is the usual Möbius fuctio Note that fo give τ CP d, if thee exists i such that ( i, d) b i the X τ 0 ad othewise we have X τ s ( i, d) l(τ) 1 i by Lemma 31 Thus by (41) we have! N(, b) ( 1) l(τ) C(τ) X τ τ C ( 1) l(τ) C(τ) X τ d τ CP d d,( i,d) b i τ CP d ( 1) l(τ) C(τ) ( i, d) l(τ) 1 i

12 JIYOU LI AND DAQING WAN Fo give j, deote by TP d(j) ad CP d(j), the umbe of pemutatios i TP d ad CP d espectively with j cycles If d, ( i, d) b i fo ay 1 i s, the we have ( 1) l(τ) C(τ) ( i, d) l(τ) 1 ( i, d) ( 1) j CP d (j) j 1 τ CP d i j1 s ( i, d) j1( 1) j µ(/d)tp (j) j 1 d s ( i, d) µ(/d) ( 1) j TP (j) j d j1 s ( i, d) µ(/d)( 1) + /! d The last equality follows fom Lemma 32 Thus we have N(, b) 1 i, d) d,( i,d) b i ( µ(/d)( 1) + / d 1 ( 1) + / ( i, d) 1 (,) ( 1) + / d,( i,d) b i µ(/d) d,( i,d) b i µ(/d) ( i, d) The last equality follows by that whe we still have ( i, d)µ(/d) 0 d,(,d) b fom the poof of Lemma 33 Whe b 0, that is b i 0 fo all 1 i s, we always have ( i, d) b i ad thus N(, 0) 1 ( 1) + / µ(/d) ( i, d) (,) Coollay 41 Let N(b) be the umbe of subsets of A sum to b Fo coveiece we coside the empty set as a subset sum to 0 Let Φ(, b) ( i, d) The we have N(b) 1 d d,( i,d) b i µ(/d), odd Φ(, b)2 / Poof The poof is simila to that of Coollay 35

COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS 13 Whe A is a elemetay abelia p-goup we have the followig simple fomula Coollay 42 Let F q be the fiite field of q elemets with chaacteistic p Let A be ay additive subgoup of F q ad A Fo ay b A, deote N(, b) to be the umbe of -subsets of A whose elemets sum to b Defie v(b) 1 if b 0, ad v(b) 1 if b 0 If p, the N(, b) 1 If p, the N(, b) 1 + ( 1) + v(b) p /p /p Poof Sice i this case we have p t ad i p, b (b 1, b 2,, b t ) It follows fom the fomula give by Theoem 11 that whe p, the (, ) 1 ad N(, b) 1 (p, d) 1 d 1,(p,d) b i µ(1/d) Whe p, assumig that 0 p t0 ad (p, 0 ) 1 we have N(, b) 1 ( 1) + / (p, d) 1 p t 0 ( ) + ( 1) + p 1 d,(p,d) b i µ(/d) ( /p /p ) d p,(p,d) b i µ(p/d) (p, d) If b 0 the b i 0 fo 1 i t the N(, 0) 1 ( ) + ( 1) + p 1 /p ( 1) /p If b 0 the thee is some i such that b i 0 ad sice (b i, p) 1 we ca oly tae d 1 i the summatio Thus N(, b) 1 ( 1) + 1 p /p /p Whe A F q, the fomula was fist foud by the authos i [5] Refeeces [1] Q Cheg, Had poblems of algebaic geomety codes, IEEE Tas & Ifom Theoy, 2008, 54(1), 402 406 [2] Q Cheg ad E Muay, O decidig deep holes of Reed-Solomo codes, I: TAMS 2007, Lectue Notes i Compute Sciece, Vol 4484, Spige, 2007 [3] TH Come, CE Leiseso, RL Rivest ad C Stei, Itoductio to Algoithms, MIT Pess ad McGaw-Hill, 2001 [4] RL Gaham, DE Kuth ad O Patashi, Cocete Mathematics: A Foudatio fo Compute Sciece, 2d ed Readig, MA: Addiso-Wesley, 1994 [5] Jiyou Li ad D Wa, O the subset sum poblem ove fiite fields, Fiite Fields & Applicatios, 14 (2008), 911 929 [6] Jiyou Li ad D Wa, A ew sieve fo distict coodiate coutig, Sciece i Chia Seies A, 53 (2010), 2351 2362

14 JIYOU LI AND DAQING WAN [7] Nicol C A, Liea cogueces ad the Vo Stemec fuctio, Due Math J, 26 (1959), 193 197 [8] AM Odlyzo ad RP Staley, Eumeatio of powe sums modulo a pime, J Numbe Theoy, 10 (1978), o 2, 263 272 [9] Ramaatha K G, Some applicatios of Ramauja s tigoometical sum C m(), Poceedigs of the Idia Academy of Scieces, vol 20 (1945), 62 69 [10] RP Staley, Eumeative combiatoics Vol 1, Cambidge Uivesity Pess, Cambidge, 1997 Depatmet of Mathematics, Shaghai Jiao Tog Uivesity, Shaghai, PR Chia E-mail addess: lijiyou@sjtueduc Depatmet of Mathematics, Uivesity of Califoia, Ivie, CA 92697-3875, USA E-mail addess: dwa@mathuciedu