momihara@educkumamoto-uacjp 16-06-2014
Definition Definition: Cayley graph G: a finite abelian group D: an inverse-closed subset of G (0 D and D = D) E := {(x, y) x, y G, x y D} (G, E) is called a Cayley graph, denoted by Cay(G, D) D is called the connection set of (G, E)
Paley graph G = Z 3 Z 3, D = {(0, 1), (0, 2), (2, 1), (1, 2)}
Definition Definition: Translation scheme Γ i := (G, E i ), 1 i d: Cayley graphs on an abelian group G R i : connection sets of (G, E i ) R 0 := {0} (G, {R i } d ) is called a translation scheme (TS) i=0 if (G, {Γ i } d ) is an association scheme (AS) i=0 In other words d R i=0 i = G, R i R j = {z (x, z) E i, (y, z) E j } is const according to l st (x, y) E l {z x z R i, y z R j } is const according to l st x y R l (R i + x y) R j is const according to l st x y R l (R i + w) R j is constant according to l st w R l
Fundamentals of characters of abelian groups A character ψ of G is a homomorphism from G to C Ĝ: the set of all characters of G e: the exponent of G Remark The image of ψ is an eth root of unity since ψ(x) e = ψ(x e ) = ψ(1 G ) = 1 Note that ψ(1 G ) = 1 by ψ(1 G ) 2 = ψ(1 G )
Fundamentals of characters of abelian groups Remark Define ψ 0 (g) := 1 for g G Then ψ 0 is a character, called the trivial character Theorem Define ψ 1 (g) := ψ(g) 1 for a character ψ Then ψ 1 is a character, called the inverse of ψ Define ψ 1 ψ 2 (g) := ψ 1 (g)ψ 2 (g) for characters ψ 1, ψ 2 Then ψ 1 ψ 2 is a character Ṭhe set Ĝ forms a group isomorphic to G
Example Example: Z 3 Possible cases: ψ((0, 1, 2)) = (1, 1, 1), (1, 1, ω), (1, 1, ω 2 ), (1, ω, 1), (1, ω 2, 1), (1, ω, ω), (1, ω, ω 2 ), (1, ω 2, ω 2 ), (1, ω 2, ω) By noting that ψ(1)ψ(2) = ψ(1 + 2) = ψ(0) = 1, Only ψ((0, 1, 2)) = (1, 1, 1), (1, ω, ω 2 ), (1, ω 2, ω) are possible These three are all characters of Z 3
Orthogonal relations Theorem (1) For ψ, ψ Ĝ, ψ(g)ψ (g) = δ ψ1,ψ 2 G, g G where δ ψ,ψ = { 1 if ψ = ψ, 0 if ψ ψ (2) For g, h G, ψ(g)ψ(h) = δ g,h G ψ G where δ g,h = { 1 if g = h, 0 if g h
Orthogonal relations Proof of (1): Put ϕ = ψψ 1 If ϕ = ψ 0, ϕ(g) = g G 1 = G g G If ϕ ψ 0, ϕ(g ) ϕ(g) = ϕ(g )ϕ(g) = ϕ(g g) = ϕ(g), g G g G which implies that g G ϕ(g) = 0 g G g G
Eigenvalues of Cayley graphs Γ: a Cayley graph on an abelian group G with connection set D Ĝ: the character group of G M: the character table of G (Each of rows and columns are labeled by the elements of Ĝ and the elements of G, respectively The (ψ, g)-entry is defined by ψ(g)) A: the adjacency matrix of Γ (Each row and column are labeled similar to the columns of M) Theorem: Eigenvalues and character sums M AM T = diag G ψ(x) x D ψ Ĝ ie, the eigenvalues of A are given by ψ(d), ψ Ĝ,
Proof (MM T ) ψ,ψ = ψ(h)ψ (h) = ψψ 1 (h) = h G h G This implies that M/ G is an orthogonal matrix By (M A) ψ,g = ψ(h) = ψ(e + g), we have (M AM T ) ψ,ψ = = = h G;h g D e D { G if ψ = ψ, 0 if ψ ψ, ψ(e + g)ψ (g) = ψ(e) ψ(g)ψ (g) g G e D ψ(e) ψψ 1 (g) e D g G e D { G e D ψ(e) if ψ = ψ, 0 if ψ ψ g G
Primitivity of translation schemes Γ: a Cayley graph on an abelian group G with connection set D Lemma Γ is connected ψ(d) D for any nontrivial ψ Ĝ Remark For any graph Γ, Γ has valency k Γ contains k as an eigenvalue Γ has valency k and is connected k occurs exactly once as an eigenvalue Γ has valency D, and all eigenvalues are given by ψ(d) For trivial ψ 0 Ĝ, ψ 0 (D) = D Γ is connected iff ψ(d) D for any nontrivial ψ Ĝ
Dual of translation schemes R 0 = {0}, R 1, R 2,, R d : an (inverse-closed) partition of G This partition induces a partition S 0 = {ψ 0 }, S 1, S 2,, S e, of Ĝ: ψ, ϕ Ĝ \ {ψ 0 } are in the same S j iff ψ(r i ) = ϕ(r i ) for 1 i d Theorem (Bridges-Mena, 1982) It holds that d e In particular, (G, {R i } d ) forms a TS iff d = e i=0 R 0 R 1 R 2 R 3 ψ 0 S 0 1 R 1 R 2 R 3 ψ S 1 1 a 1 a 2 a 3 ψ S 2 1 b 1 b 2 b 3 ψ S 3 1 c 1 c 2 c 3 If (G, {R i } d ) forms a TS, then so does (Ĝ, {S i } d ), which is called i=0 i=0 the dual of (G, {R i } d ) i=0 G P 1 is the first eigenmatrix of (Ĝ, {S i } d ) i=0 for the first eigenmatrix P of (G, {R i } d i=0 )
Cyclotomic scheme F q : the finite field of order q F q : the multiplicative group of F q C : F q st C = C Lemma: Cyclotomic scheme The partition F q /C of F q gives a TS on (F q, +), called a cyclotomic scheme Each coset (called a cyclotomic coset) of F q /C is expressed as C (N,q) i = γ i γ N, 0 i N 1, where N q 1 is a positive integer and γ is a fixed primitive element of F q For w C (N,q), l p l i, j = (C(N,q) i Hence, p l is depending on l not w i, j + w) C (N,q) = (C (N,q) + 1) C (N,q) j i l j l
Characters of finite fields There are two kinds of characters for finite fields, which are additive characters and multiplicative characters Lemma For a fixed primitive element γ F q, χ j : F q C, 0 j q 2, defined by χ j (γ k ) := ζ jk q 1 are all multiplicative characters of F q, where ζ q 1 = e 2πi Define the trace Tr q m /q from F q m to F q by Tr q m /q(x) = x + x q + x q2 + + x qm 1, which is a homomorphism from (F q m, +) to (F q, +) q 1
Characters of finite fields Lemma The function ψ j : F q C, j F q, defined by are all additive characters of F q ψ j (x) = ζ Tr q/p( jx) p It holds that ψ j (x + y) = ψ j (x)ψ j (y) since Tr is a homomorphism from F q to F p ψ 1 is called canonical Note that ψ a (x) = ψ 1 (ax) and ψ(x) = ψ( x)
Gauss sums and Jacobi sums Definition For the canonical additive character ψ of F q and a nontrivial multiplicative character χ of F q, the sum G(χ) := ψ(x)χ(x) is called a Gauss sum x F q Definition For two multiplicative characters χ 1, χ 2 of F q, the sum J(χ 1, χ 2 ) = χ 1 (1 x)χ 2 (x) is called a Jacobi sum x F q \{0,1}
Basic properties of Gauss sums and Jacobi sums Lemma For any nontrivial multiplicative characters χ 1, χ 2 of F q st χ 1 χ 2 is nontrivial, then J(χ 1, χ 2 ) = G(χ 1)G(χ 2 ) (1) G(χ 1 χ 2 ) Lemma For any nontrivial multiplicative character χ of F q, G(χ)G(χ) = q The lemma above implies that G(χ) = q Furthermore, by (1), we have J(χ 1, χ 2 ) = q
G(χ)G(χ) = q Proof: G(χ)G(χ) = ψ 1 (x)ψ 1 ( y)χ(x)χ 1 (y) = x,y F q ψ 1 (x y)χ(xy 1 ) (2) x,y F q Write z = xy 1 Then, (2) = χ(z)ψ 1 (y(z 1)) = y,z F q χ(z) ψ 1 (y(z 1)) χ(z) = q y F q z F q z F q
Intersection numbers of cyclotomic schemes Definition (C (N,q) + 1) C (N,q), 0 i, j N 1, are called cyclotomic i numbers, denoted by (i, j) N j Computation of (i, j) N : The characteristic function of C (N,q) on F i q is given by f i (γ a ) = 1 N N 1 k=0 ζ ik N χk (γ a ), where χ is a multiplicative character of order N of F q st χ(γ a ) = ζ a Then, we have N (i, j) N = f j (x) f i (x 1) (3) x F q \{0,1}
Intersection numbers of cyclotomic schemes p l i, j (3) = = = 1 N 1 N 2 x F q \{0,1} k,l=0 1 N 1 N 2 k,l=0 1 N 1 N 2 k,l=0 ζ ζ (ik+ jl) N (ik+ jl) N ζ (ik+ jl) N χ l ( 1) J(χ k, χ l ) χ k (x)χ l (x 1) x F q \{0,1} χ k (x)χ l ( x + 1) could be expressed as a linear combination of Jacobi sums! Remark Since J(χ k, χ l ) = q for k, l, k + l 0 (mod N), (i, j) N q 3N + 1 N 2 (N2 3N + 2) q N 2
Eigenvalues of cyclotomic schemes The eigenvalues are given by ψ(c (N,q) ), ψ Ĝ, called Gauss i periods We can write ψ(x) = ψ 1 (ax) for some a F q, where ψ 1 is the canonical additive character of F q Thus, ψ(c (N,q) ) = ψ i 1 (C (N,q) ), where a C (N,q) i+l l Write η i = ψ 1 (C (N,q) ) Then, the first eigenmatrix of the cyclotomic i scheme is given by 1 q 1 N q 1 N q 1 q 1 N N 1 η 0 η 1 η 2 η N 1 1 η 1 η 2 η 3 η 0 1 η N 1 η 0 η 1 η N 2
Gauss periods and Gauss sums Lemma: Gauss periods and Gauss sums (i) ψ 1 (C (N,q) ) = 1 N 1 G(χ h )χ h (γ i ), i N (ii) G(χ) = N 1 h=0 where χ is a multiplicative character of order N of F q i=0 ψ 1 (C (N,q) )χ(γ i ), i
Eigenvalues of cyclotomic schemes ψ 1 (C (N,q) ) = 1 i N = 1 N = = = 1 N ψ 1 (γ i x N ) x F q x F q 1 (q 1)N 1 (q 1)N χ C 0 1 q 1 where C 0 is the subgroup of F q ψ 1 (y) χ(γ i x N )χ(y) y F q x F q χ F q χ F q G(χ 1 )χ(γ i x N ) G(χ 1 )χ(γ i ) χ(x N ) χ F q G(χ 1 )χ(γ i ), x F q consisting of all χ trivial on C(N,q) 0
Evaluated Gauss sums (Small order) Gauss sums of order N 24 have been partially evaluated (Berndt et al, 1997) (Pureness) When Gauss sums take the form ζ N q was determined (Aoki, 2004, 2012) In particular, if 1 p (mod N), then G(χ N ) takes a rational value (Baumert et al, 1982) (Index 2 or 4 case) In the case where [Z : p ] = 2, Gauss N sums have been completely evaluated (Yang et al, 2010) In the case where [Z : p ] = 4, Gauss sums have been N partially evaluated (Feng et al, 2005)
Useful formulas on Gauss sums Hasse-Davenport product formula η: a mult character of order l > 1 of F p f For nontrivial mult character χ of F p f, G(χ) = G(χl ) χ l (l) l 1 i=1 G(η i ) G(χη i ) Hasse-Davenport lifting formula χ : a nontrivial mult character of F q χ: the lift of χ to F q m, ie, χ (α) = χ(α qm 1 q 1 ) for α F q m Then G q m(χ) = ( 1) m 1 (G q (χ )) m
Useful formulas on Gauss sums Stickelberger s formula N: a positive integer p: a prime st gcd (p, N) = 1 f: the order of p in Z N O M : the rings of integers of M = Q(ζ N, ζ p ) p: a prime ideal of O M lying over p σ j : Gal(M/Q(ζ p )) by σ j (ζ N ) = ζ j N, j Z N T := Z / p N Then, it holds that G f (χ 1 )O M = p t T s p ( t(q 1) )σ 1 k t, where s p ( t(q 1) N ) is the sum of all a i for t(q 1) N = n i=0 a i p i
Useful formulas on Gauss sums A useful algorithm for computing the p-divisibility of Gauss sums was found by Helleseth et al, 2009, called the modular p-ary add-with-carry algorithm A generalization of Stickelberger s formula (congruence) in p-adic fields was found by Gross and Koblitz, 1979
Cyclotomic schemes, cyclic codes, and related geometry Remarks The computation of eigenvalues of cyclotomic schemes is equivalent to that of weight distributions of certain cyclic codes, called irreducible cyclic codes A strongly regular graph obtained as a fusion of a cyclotomic scheme is described in terms of projective geometry
Irreducible cyclic codes Definition: Irreducible cyclic code f(x): an irreducible divisor of x m 1 F p [x], where gcd (m, p) = 1 The cyclic code of length m over F p generated by (x m 1)/ f(x) is called an irreducible cyclic code (This code has no proper cyclic subcodes) f: the order of p modulo m q := p f = 1 + km γ: a primitive root of F q f(x) := f 1 i=0 (x γkpi ) F p [x] irreducible over F p g(x) := l S(x γ kl ), where S = {l 0 l m 1, l a power of p (mod m)}
Irreducible cyclic codes Lemma C: the cyclic code generated by g(x) The q codewords in C are given by h α (x) := (Tr(α), Tr(αγ k ), Tr(αγ 2k ),, Tr(αγ (m 1)k )), α F q Proof: For any l S, h α (γ kl ) = h α (x) := m 1 j=0 m 1 j=0 Tr(αγ jk )x j, α F q f 1 Tr q/p (αγ jk )γ kl j = Hence, g(x) h α (x), ie, h α (x) C i=0 m 1 α pi j=0 γ jk(l pi) = 0
Proof Since C = q, it remains to show that h α (x) are all distinct Assume h α (x) = h β (x) Then, for ω := α β F q Tr(ω) = Tr(ωγ k ) = Tr(ωγ 2k ) = = Tr(ωγ (m 1)k ) = 0 For any choice of a j F p, 0 = f 1 j=0 f 1 a j Tr(ωγ jk ) = Tr(ω a j γ jk ) Since {1, γ k,, γ ( f 1)k } is a basis of F q over F p, the above is impossible j=0
Weight-distribution Theorem (McEliece) Let N := gcd (k, (q 1)/(p 1)) Then, w(h α (x)) = m(p 1) p 1 p pk ψ 1(αC (N,pf ) ) 0 Proof: Let χ be a mult character of order k of F q w(h α (x)) =m 1 p m 1 i=0 x F p ψ 1 (xαγ ki ) m(p 1) = 1 m 1 p p ψ 1 (xαγ ki ) i=0 x F p m(p 1) = 1 k 1 p pk G(χ j )χ j (xα) j=0 x F p
Weight-distribution Since for any y F p χ j (x) = χ j (yx) = χ j (y) χ j (x), x F p x F p x F p x F p χ j (x) = 0 iff χ j is nontrivial on F p Let χ be a mult character of order N of F q Then, m(p 1) w(h α (x)) = p 1 N 1 G(χ j )χ j (α) p pk j=0 m(p 1) = p 1 p pk ψ 1(αC (N,pf ) ) 0 Problem Characterize all two or three weight irreducible cyclic codes
Fusion schemes Given a d-class AS (X, {R i } d ), we can take union of classes to i=0 form graphs with larger edge sets (this process is called a fusion) Problem Given an N-class cyclotomic scheme on F q, determine its fusion schemes X j, j = 1, 2,, d: a partition of Z N The Bridges-Mena theorem (more generally, the Bannai-Muzychuk criterion) implies that i X j C (N,q) s forms a TS i iff a partition Y h, h = 1, 2,, d, of Z N st each ψ(γ a i X j C (N,q) ) is const according to a Y i h
Gauss sum and trace zero W consider 2-class fusion schemes (strongly regular graphs) of cyclotomic schemes of order N = qm 1 q 1 Proposition Let χ be a mult character of order N of F q m Let S 0 := {log γ x (mod N) Tr q m /q(x) = 0, x 0} Then, G(χ) = q χ(ω i ) i S 0 L :=a system of representatives of F q m/f q G(χ) = a F x L q χ(xa)ζ Tr q m /p(xa) p = =(q 1) χ(γ i ) i S 0 χ(x) x L a F q χ(γ i ) = q χ(γ i ) i L\S 0 i S 0 ζ Tr q/p(atr q m /q (x)) p
Geometric understanding X: a subset of Z N When is Γ = Cay( i X C (N,qm ) ) strongly regular? i (Γ is strongly regular iff ψ(γ a i X C (N,qm ) ), a = 0, 1,, N 1, i take exactly two values) ψ(γ a i X C (N,qm ) ) = 1 i N = q N i X χ G(χ 1 )χ(γ a+i ) X χ χ 0 N i X =q X (S 0 a) X, χ(γ a+i j ) X (1 + q S 0 ) N j S 0 where χ ranges through all mult characters of exponent N of F q m
Geometric understanding Proposition (Delsarte, 1972) Cay( i X C (N,qm ) ) is strongly regular iff X (S i 0 a), a Z N, take exactly two values Note that each S 0 a is a hyperplane of PG(m 1, q) Problem Find a subset X of PG(m 1, q), which has two intersection numbers with the hyperplanes of PG(m 1, q) (X is called a two-intersection set in PG(m 1, q)) See Caldervbank-Kantor (1986) for more on the geometric aspect of strongly regular graphs on F q
References Gauss sums and related character sums 1 B Berndt, R Evans, KS Williams, Gauss and Jacobi Sums, 1997 2 K Ireland, M Rosen, A Classical Introduction to Modern Number Theory, 2nd ed, 2003 3 R Lidl, H Niederreiter, Finite Fields, 1997 4 WM Schmidt, Equations over Finite Fields: An Elementary Approach, 2nd ed, 2004 5 A Robert, A Course in p-adic Analysis, 2000 Linkage with difference sets 1 T Storer, Cyclotomy and Difference Sets, 1967 2 RJ Turyn, Character sums and difference sets, Pacific J Math, (1965)
References Linkage with codes 1 AR Calderbank, WK Kantor, The geometry of two-weight codes, Bull London Math Soc, (1986) 2 B Schmidt, C White, All two-weight irreducible cyclic codes?, Finite Fields Appl, (2002) Computations of Gauss sums 1 LD Baumert, WH Mills, RL Ward, Uniform cyclotomy, J Number Theory, (1982) 2 KQ Feng, J Yang, SX Luo, Gauss sums of index 4: (1) cyclic case, Acta Math Sin, (2005) 3 J Yang, SX Luo, KQ Feng, Gauss sums of index 4: (2) non-cyclic case, Acta Math Sin, (2006) 4 J Yang, L Xia, Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci China Ser A, (2010) 5 N Aoki, On pure Gauss sums, Com Math Univ Sancti Pauli, (2012)
References General theory 1 AE Brouwer, WH Haemers, Spectra of Graphs, 2012 2 A Terras, Fourier Analysis on Finite Groups and Applications, 1999 p-divisibility: Modular p-ary add-with-carry algorithm 1 T Helleseth, HDL Hollmann, A Kholosha, Z Wang, Q Xiang, Proofs of two conjectures on ternary weakly regular bent functions, IEEE Trans Inform Theory, (2009) Japanese book 1,,,, 2001 2 - -,,, 2003