Some remarks for codes and lattices over imaginary quadratic

Transcription

1 Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. Abstract Let l > 0 be a square-free iteger, l 3 mod, K = Q( l), ad O K the rig of itegers of K. Codes C over rigs R := O K/pO K determie lattices Λ l (C) over K. The theta series θ Λl (C) of such lattice ca be writte i terms of the complete weight eumerator of C. For ay l > l the first l+1 terms of their correspodig theta fuctios are the same with those of Λ l (C). I [6] it was cojectured that for l > p(+1)(+2) there is a uique complete weight 2 eumerator correspodig to a give theta fuctio. I this paper, we explore this cojecture ad some ew computatioal results. 1 Itroductio Keywords Codes, theta fuctios, complete weight eumerators Let l > 0 be a square-free iteger cogruet to 3 modulo, K = Q( l) be the imagiary quadratic field, ad O K its rig of itegers. Codes, Hermitia lattices, ad their theta-fuctios over rigs R := O K /po K, for small primes p, have bee studied by may authors, see [1], [], [5], amog others. I [1], explicit descriptios of theta fuctios ad MacWilliams idetities are give for p = 2, 3. I [7] we explored codes C defied over R for p > 2. For ay l oe ca costruct a lattice Λ l (C) via Costructio A ad defie theta fuctios based o the structure of the rig R. Such costructios suggested some relatios betwee the complete weight eumerator of the code ad the theta fuctio of the correspodig lattice. I this paper we further study the weight eumerators of such codes i terms of the theta fuctios of the correspodig lattices. For ay prime p with p l, let R := O K /po K = { a + bω : a, b F p, ω 2 + ω + d = 0 }, where d = (l + 1)/. We have the map ρ l,p : O K O K /po k =: R A liear code C of legth over R is a R-submodule of R. The dual is defied as C = {u R : u v = 0 for all v C}. If C = C the C is self-dual. We defie Λ l (C) := {u = (u 1,..., u ) O K : (ρ l,p (u 1 ),..., ρ l,p (u )) C}, I other words, Λ l (C) cosists of all vectors i OK i the iverse image of C, take compoetwise by ρ l,p. This method of lattice costructio is kow as Costructio A. Let τ H = {z C : Im(z) > 0}, the upper-half plae. Ad let q = e πiτ. For ay lattice Λ i K, we have a associated theta fuctio θ Λ (q), give by θ Λ (q) = z Λ q z z, where deotes the usual dot product ad z deotes compoet-wise cojugatio. Thus, for ay liear code C, we have a associated lattice Λ l (C) ad associated theta fuctio θ Λl (C)(q).

2 For otatio, let r a+pb+1 = a bω, so R = { r 1,..., r p 2}. For a codeword u = (u1,..., u ) R ad r i R, we defie the coutig fuctio i (u) := #{j : u j = r i }. The complete weight eumerator of the R code C is the polyomial cwe C (z 1, z 2,..., z p 2) = u C z 1(u) 1 z 2(u) 2... z p 2 (u) p 2. (1) We ca use the complete weight eumerator polyomial to fid the theta fuctio of the lattice Λ l (C). For a proof of the followig result see [7]. Lemma 1. Let C be a code defied over R ad cwe C its complete weight eumerator as above. For itegers a ad b ad a prime p, let Λ a,b deote the lattice a bω l + po K. The, θ Λl (C)(q) = cwe C (θ Λ0,0 (q), θ Λ1,0 (q),..., θ Λp 1,p 1 (q)). Note that the q 2 argumets of this polyomial ca be computed i terms of certai oedimesioal theta series which are defied i Sectio 2.1 of [7]. I [2], for p = 2, the symmetric weight eumerator polyomial swe C of a code C over a rig or field of cardiality is defied to be swe C (X, Y, Z) = cwe C (X, Y, Z, Z). For Λ Λl (C)(q), the lattice obtaied from C by Costructio A, by Theorem 5.2 of [2], oe ca the write θ Λl (C)(q) = swe C (θ Λ0,0 (q), θ Λ1,0 (q), θ Λ0,1 (q)). These theta fuctios are referred to as A d (q), C d (q), ad G d (q) i [2] ad [8]. Remark 1. The coectio betwee complete weight eumerators of self-dual codes over F p ad Siegel theta series of uimodular lattices is well kow. Costructio A associates to ay legth code C = C a -dimesioal uimodular lattice; see [3] for details. For p > 2, however, there are (p+1)2 theta fuctios associated to the various lattices, so our aalog of the symmetric weight eumerator polyomial eeds more tha 3 variables. Problem 1. Determie a explicit relatio betwee theta fuctios ad the symmetric weight eumerator polyomial of a code defied over R for p > 3. We expect that the aswer to the above problem is that the theta fuctio is give as the symmetric weight eumerator swe C of C, evaluated o the theta fuctios defied o cosets of O K /po K. 2 Theta fuctios ad the correspodig complete weight eumerator polyomials For a fixed prime p, let C be a liear code over R = F p 2 or F p F p of legth ad dimesio k. A admissible level l is a iteger l such that O K /po K is isomorphic to R. For a admissible l, let Λ l (C) be the correspodig lattice as i the previous sectio. The, the level l theta fuctio θ Λl (C)(q) of the lattice Λ l (C) is determied by the complete weight eumerator cwe C of C, evaluated o the theta fuctios defied o cosets of O K /po K. We cosider the followig questios. How do the theta fuctios θ Λl (C)(q) of the same code C differ for differet levels l? Ca o-equivalet codes give the same theta fuctios for all levels l? We give a satisfactory aswer to the first questio (cf. Theorem 1, Lemma 2) ad for the secod questio we cojecture that: Cojecture 1. Let C be a code of size defied over R ad θ Λl (C) be its correspodig theta fuctio for level l. The, for large eough l, there is a uique complete weight eumerator polyomial which correspods to θ Λl (C). Let C be a code defied over R for a fixed p > 2. Let the complete weight eumerator of C be the degree polyomial cwe C = f(x 1,..., x r ), for r = p 2. The from Lemma 1 we have that θ Λl (C)(q) = f(θ Λ0,0 (q),..., θ Λp 1,p 1 (q)) for a give l. First we wat to address how θ Λl (C)(q) ad θ Λl (C)(q) differ for differet l ad l.

3 Theorem 1. Let C be a code defied over R. For all admissible l, l with l < l the followig holds θ Λl (C)(q) = θ Λl (C)(q) + O(q l+1 ). Proof. See [6] for details. We have the followig lemma; see [6]. Lemma 2. Let C be a fixed code of size defied over R ad θ(q) = λ i q i be its theta fuctio for level l. The, there exists a boud B l,p, such that θ(q) is uiquely determied by its first B l,p, coefficiets. For otatio, whe p ad are fixed, we will let B l = B l,p,. To exted the theory for p = 2 to p > 2 we have to fid a relatio betwee the theta fuctio θ Λl (C) ad the umber of complete weight eumerator polyomials correspodig to it. Fix a odd prime p ad let C be a give code of legth over R. Choose a admissible value of l such that there are (p+1)2 idepedet theta fuctios. The, the complete weight eumerator of C has degree ad r = (p+1)2 variables x 1,..., x r. We call a geeric complete weight eumerator polyomial a homogeous polyomial P Q[x 1,..., x r ]. Deote by P (x 1,..., x r ) a geeric r-ary, degree, homogeeous polyomial. Assume that there is a legth code C defied over R such that P (x 1,..., x r ) is the symmetric weight eumerator polyomial. I other words, Fix the level l. The, by replacig swe C (x 1,..., x r ) = P (x 1,..., x r ) x 1 = θ Λ0,0 (q),......, x r = θ Λp 1,p 1 (q), we compute the left side of the above equatio as a series i=0 λ iq i. By equatig both sides of i=0 λ iq i = P (x 1,..., x r ), we ca get a liear system of equatios. Sice the first λ 0,..., λ Bl 1 determie all the coefficiets of the theta series the we have to pick B l equatios (these equatios are ot ecessarily idepedet). Cosider the coefficiets of the polyomial P (x 1,..., x r ) as parameters c 1,... c s. The, the liear map L l : C s C B l 1 (c 1,... c s ) (λ 0,..., λ Bl 1) has a associated matrix M l. For a fixed value of (λ 0,..., λ Bl 1), determiig the rak of the matrix M l would determie the umber of polyomials givig the same theta series. There is a uique complete weight eumerator correspodig to a give theta fuctio whe ull (M l ) = s rak (M l ) = 0 Cojecture 2. For l p(+1)(+2) 1 we have ull M l = 0, or i other words ) ( 1 + (p+1)2! rak (M l ) = ( )! (p+1) 2 1! The choice of l is take from experimetal results for primes p = 2 ad 3. More details are give i the ext sectio. It is obvious that Cojecture 2 implies Cojecture 1. If Cojecture 1 is true the for large eough l there would be a oe to oe correspodece betwee the complete weight eumerator polyomials ad the correspodig theta fuctios. Perhaps, more iterestig is to fid l ad p for which there is ot a oe to oe such correspodece. Cosider the map Φ(l, p) = (λ 0 (l, p),..., λ Bl 1(l, p)), where λ 0,..., λ Bl 1 are ow fuctios i l ad p. Let V be the variety give by the Jacobia of the map Φ. Fidig iteger poits l, p o this variety such that l ad p satisfy our assumptios would give us values for l, p whe the above correspodece is ot oe to oe. However, it seems quite hard to get explicit descriptio of the map Φ. Next, we will try to shed some light over the above cojectures for fixed small primes p.

4 3 Bouds for small primes I [8] we determie explicit bouds for the above theorems for prime p = 2. I this sectio we give some computatio evidece for the geeralizatio of the result for p = 3. We recall the theorem for p = 2. Theorem 2 ([8], Thm. 2). Let p = 2 ad C be a code of size defied over R ad θ Λl (C) be its correspodig theta fuctio for level l. The the followig hold: i) For l < 2(+1)(+2) 1 there is a δ-dimesioal family of symmetrized weight eumerator polyomials correspodig to θ Λl (C), where δ (+1)(+2) 2 (l+1) 1. ii) For l 2(+1)(+2) 1 ad < l+1 there is a uique symmetrized weight eumerator polyomial which correspods to θ Λl (C). These results were obtaied by usig the explicit expressio of theta i terms of the symmetric weight eumerator valuated o the theta fuctios of the cosets. Next we wat to fid explicit bouds for p = 3 as i the case of p = 2. I the case of p = 3 it is eough to cosider four theta fuctios, θ Λ0,0 (q), θ Λ1,0 (q), θ Λ0,1 (q), ad θ Λ1,1 (q) sice θ Λ2,0 (q) = θ Λ1,0 (q), θ Λ2,2 (q) = θ Λ1,1 (q) ad θ Λ0,2 (q) = θ Λ1,2 (q) = θ Λ2,1 (q) = θ Λ0,1 (q). If we are give a code C ad its weight eumerator polyomial the we ca fid the theta fuctio of the lattice costructed from C usig Costructio A. Let θ(q) = i=0 λ iq i be the theta series for level l ad p(x, y, z, w) = c i,j,k x i y j z k w m i+j+k+m= be a degree geeric -ary homogeeous polyomial. We would like to fid out how may polyomials p(x, y, z, w) correspod to θ(q) for a fixed l. For a give l fid θ Λ0,0 (q), θ Λ1,0 (q), θ Λ0,1 (q) ad θ Λ1,1 (q) ad substitute them i the p(x, y, z, w). Hece, p(x, y, z, w) is ow writte as a series i q. We get ifiitely may equatios by equatig the correspodig coefficiets of the two sides of the equatio p(θ Λ0,0 (q), θ Λ1,0 (q), θ Λ0,1 (q), θ Λ1,1 (q)) = λ i q i. Sice the first λ 0,..., λ Bl 1 determie all the coefficiets of the theta series the it is eough to pick the first B l equatios. The liear map L l : (c 1,... c 20 ) (λ 0,..., λ Bl 1) has a associated matrix M l. If the ullity of M l is zero the we have a uique polyomial that correspods to the give theta series. We have calculated the ullity of the matrix ad B l for small ad l. Example 1 (The case p = 3, = 3). The geeric homogeous polyomial is give by i=0 P (x, y, z) = c 1 x 3 + c 2 x 2 y + c 3 x 2 z + c x 2 w + c 5 xy 2 + c 6 xz 2 + c 7 xw 2 + c 8 xyz + c 9 xyw + c 10 xzw + c 11 y 3 + c 12 y 2 z + c 13 y 2 w + c 1 yz 2 + c 15 yw 2 + c 16 yzw + c 17 z 3 + c 18 z 2 w + c 19 zw 2 + c 20 w 3. (2) The system of equatios ca be writte by the form of A c = λ where c = ( c 1 c 2 c 20 ) t, λ = ( λ0 λ 1 λ 15 ) t. I the case of l = 7 the matrix M7 has ull (M 7 ) =. We have a positive dimesio family of solutio set. The case of l = 11 the matrix M 11 has ull (M 11 ) = 1. For ay case where l 19 the ullity of the matrix is 0. Hece, for every give theta series, there is a uique symmetric weight eumerator polyomial.. We summarize the results i the followig table:

5 l = 3 = = 5 B l ull M l B l ull M l B l ull M l Recall that l 3 mod ad p l. It seems from the table that the same boud of B l = 2(+1)(+2) as for p = 2 holds also for p = 3, = 3. We have the followig cojecture for geeral p, ad l. Cojecture 3. For a give theta fuctio θ Λl (C) of a code C for level l there is a uique complete weight eumerator polyomial correspodig to θ Λl (C) if l p(+1)(+2). It is iterestig to cosider such questio for such lattices idepedetly of the coectio to codig theory. What is the meaig of the boud B l for the rig O K /po K? Do the theta fuctios defied here correspod to ay modular forms? Is there ay differece betwee the cases whe the rig is F p F p or F p 2? Refereces [1] C. Bachoc, Applicatios of codig theory to the costructio of modular lattices. J. Combi. Theory Ser. A 78 (1997), o. 1, [2] K. S. Chua, Codes over GF() ad F 2 F 2 ad Hermitia lattices over imagiary quadratic fields. Proc. Amer. Math. Soc. 133 (2005), o. 3, (electroic). [3] J. Leech ad N. J. A. Sloae, Sphere packig ad error-correctig codes, Caadia J. Math., 23, (1971), [] F. J. MacWilliams, N. J. A. Sloae, The theory of error-correctig codes. II. North-Hollad Mathematical Library, Vol. 16. North-Hollad Publishig Co., Amsterdam-New York-Oxford, pp. i ix ad [5] F. J. MacWilliams, N. J. A. Sloae, The theory of error-correctig codes. I. North-Hollad Mathematical Library, Vol. 16. North-Hollad Publishig Co., Amsterdam-New York-Oxford, pp. i xv ad [6] T. Shaska, C. Shor, C. S. Wijesiri, Codes over rigs of size p 2 ad lattices over imagiary quadratic fields. Fiite Fields Appl. 16 (2010), o. 2, [7] T. Shaska ad C. Shor, Codes over F p 2 ad F p F p, lattices, ad correspodig theta fuctios. Advaces i Codig Theory ad Cryptology, vol 3. (2007), pg [8] T. Shaska ad S. Wijesiri, Codes over rigs of size four, Hermitia lattices, ad correspodig theta fuctios, Proc. Amer. Math. Soc., 136 (2008), [9] T. Shaska ad C. Shor, Theta fuctios ad complete weight eumerators for codes over imagiary quadratic fields, (work i progress).