Ring of the weight enumerators of d + n

Transcription

1 Ring of the weight enumerators of Makoto Fujii Manabu Oura Abstract We show that the ring of the weight enumerators of a self-dual doubly even code in arbitrary genus is finitely generated Indeed enough elements to generate it are given The latter result is applied to obtain a minimal set of generators of the ring in genus two 1 Introduction The weight enumerator plays an important role in coding theory and has connections with other branches in mathematics We recall some of them to see the background of this paper Let C be a self-dual doubly even Type II, for short) code of length n The weight enumerator W C x, y) = x n wtv) y wtv) v C has invariant properties The so-called MacWilliams identity is described as W C x, y) = W C x + y, x y ) and the doubly evenness gives W C x, y) = W C x, 1y) These being said, the weight enumerator of a Type II code is an element of the invariant ring C[x, y] G = {fx, y) C[x, y] : σf = f σ G} of the finite group G where G is of order 19 generated by ) 1 1 1, 1 1 ) Keywords: code, weight enumerator 010MSC: Primary 94B05, Secondary 05E15 Running Head: Ring of the weight enumerators of 1

2 and ) a b σfx, y) = fax + by, cx + dy), σ = c d A remarkable fact is that the converse is also true, that is, Gleason [3] showed that the invariant ring C[x, y] G is generated by the weight enumerators of Type II codes Indeed we have C[x, y] G = C[W d + x, y), W d + x, y)] 4 We shall describe some consequences of this equality Since the degrees of the generators are and 4, length of a Type II code is a multiple of Non-existence of an extremal Type II code for sufficiently large n also follows from the above equality We observe that W d + x, y) and W d + x, y) are algebraically independent over C A finite 4 group having such a property ie, whose invariant ring is generated by the algebraically independent elements over C) is called a finite unitary reflection group See [13] The generalization of the above correspondence is investigated in, for example, [1, 7] The invariance property of the weight enumerator gives rise to the relation with the modular forms See [1,, 1] In fact, the weight enumerator of a Type II code of length n is mapped under the theta map to the Siegel modular form of weight n/ in genus g The modular form of weight which is obtained from the difference ψ g) of the weight enumerators of d+ and d + 16 is of great importance We just mention two points in genus three Witt [14] asked if the modular form obtained from ψ 3) vanishes, and it was affirmatively answered in [5, 6] Runge [10] showed that the ring of Siegel modular forms for Γ 3 is isomorphic to the quotient ring of the invariant ring of some finite group divided by an ideal ψ 3) ) Let D g) be the ring of the weight enumerator of in genus g This is a subring of the ring of the weight enumerators of Type II codes As indicated above, D 1) coincides with the ring of the weight enumerators In this paper, we show that D g) is generated by the elements of n g+3 Using this result, we show that D ) is minimally generated by nine weight enumerators of lengths, 4, 3, 40, 4, 56, 64, 7, 0 Preliminaries Let F = {0, 1} be the field of two elements Two vector spaces F n and F g appear in the following For technical reason, an element of F n is regarded as a row vector, while that of F g as a column vector The space Fn is equipped with the inner product u v = u 1 v u n v n, u = u 1,, u n ), v = v 1,, v n )

3 and so is F g α = α α g g, α = α 1 α g, = 1 g Since we deal with only binary linear codes in this paper, we call a subspace of F n a code of length n The weight wtu) of an element u F n is the number of non-zero coordinates of u A code C is said to be self-dual if it coincides with its dual C = {v F n : u v = 0, u C}, doubly even if wtu) 0 mod 4), u C Codes with those two properties self-duality and doubly evenness) are particularly interesting We use the term Type II instead of self-dual and doubly even It is known that a Type II code of length n exists if and only if n 0 mod ) The weight enumerator of a code C of length n in genus g is C x α : α F g ) = x u x u 1 =u 11,,u 1n ) C 11 u 1 u g1 u g u g =u g1,,u gn ) C x u 1n u gn This definition is consistent with that in the previous section if we put x = x 0, y = x 1 Since there does not occur any confusion, we shall use an abridged notation C It is clear that is a homogeneous polynomial of total degree n in C[x α : α F g ] Let Wg) be the ring C over C generated by the weight enumerators of all Type II codes in genus g It is known that is the invariant ring of the specified finite group cf [3, 1, 7]) In particular is finitely generated In Introduction we discussed this topic for g = 1 Next we recall a Type II code of length n for n 0 mod ) and its weight enumerator It is nice to start with a repetition code R n of length n The dual code of R n can be described as R n = {u 1,, u n ) F n : u u n = 0} which has a generator matrix The following n/ n matrix is a generator matrix of, that is, the n/ row vectors form a 3

4 basis of : The code is then characterized as = {α 1 + γ, α 1, α + γ, α,, α n/ + γ, α n/ ) : α 1,, α n/, γ F, α α n/ = 0} It is known to be Type II The weight enumerator of in genus g is expressed as = 1 g,γ F g 1) α x α+γ x α α F g We can find this formula of genus two in [] For the completeness of this paper, we add a proof We have RHS = 1 g γ F g = 1 g γ F g n/ 1) αi x αi +γx α i F g i=1 α i F g α 1,,α n/ F g F g For a fixed γ, we divide the summation as From the α 1 + +α n/ =0 α 1,,α n/ F g F g -part, we get g n/ 1) α1 + +α n/ ) x α1 +γx α 1 x α n/ +γx α n/ = α 1,,α n/ F g α 1 + +α n/ =0 α 1 + +α n/ =0 + α 1 + +α n/ 0 x α1 +γx α 1 x α n/ +γx α n/ because of 1) α1 + +α n/ ) = 1 for any F g Next fix α1,, α n/ such that α α n/ 0 Then the number of F g which is orthogonal to α1 + +α n/ 0) is g 1 This could be easily understood if you consider the dual code of a code generated by α 1 + +α n/ 4

5 n dim n dim D ) n Table: Dimensions of n and D ) n At any rate, from the α 1 + +α n/ 0 γ F g -part, we get 0 Finally we have that α 1,,α n/ F g α 1 + +α n/ =0 x α1 +γx α 1 x α n/ +γx α n/ In view of the characterization of this is nothing else but the definition of the weight enumerator of in genus g This proves the formula We denote by D g) the ring generated over C by the weight enumerators of n =, 16, 4, ) in genus g The ring D g) is a subring of These rings are graded as = D g) = n 0 mod ) n 0 mod ) n, D g) n In [], is determined Let be the Golay code of length 4 It is then = C[,,, d + 40 and the dimension formula is given as follows: n dim n = ] C[,, 1 + t 3 1 t )1 t 4 ) 1 t 40 ) The dimensions of small n in genus are given in Table Finally we recall the following Φ-operator Φxα 0 ) = x α and Φxα 1 ) = 0,, ) α F g 1 α, F g d + 40 ] d + 3 It is known that the Φ-operator maps the weight enumerator of a code in genus g to that in genus g 1 3 Results Our first objective is to prove 5

6 Theorem 1 1) D g) is finitely generated over C ) A set of generators of D g) can be obtained from for n g+3 Proof Since 1) follows from ), we shall show ) If we ignore the coefficient 1, the weight g enumerator of has the form X n/ 1 + X n/ + + X n/ g for fixed X i C[x α : α F g ] For a better understanding, we put Z i = Xi 4 Here we remind that n 0 mod ) Then we can say that D g) is generated by the forms Z 1 + Z + + Z g, Z 1 + Z + + Z g, If we apply the fundamental theorem of symmetric polynomials, we can conclude that D g) can be generated by Z i 1 + Z i + + Z i g, 1 i g Translating this into the condition of the length n, we have 1 n 1 4 g Hence, in order to generate D ), it is enough for n to range from through g+3 mod This completes the proof We shall examine the case g = 1 lengths, 16, 4, 3 to generated D 1) Because of we get W 1) ) = W 1) d + 16 From ) of Theorem 1, we possess four elements of and W 1) = 5 d D 1) = C[W 1) W 1), W 1) ] ) W 1) W 1), Notice that our argument in this section does not give guarantee as to the fact W 1) = D 1) We proceed to the higher genus Table shows that D ) is strictly smaller than In fact, we shall show Proposition We have that = D g) if and only if g = 1 6

7 Proof We have only to prove D g) for all g We know that suppose that = a ) 3 + bw g) d c for some g 3 If we successively apply the Φ-operator to both sides, we get = a We have thus a contradiction to the fact We turn our attention to the case g = ) 3 + bw ) d c / D ) This completes the proof / D ) Now Theorem 3 The ring D ) is minimally generated by nine elements of lengths, 4, 3, 40, 4, 56, 64, 7, 0 Proof By ) of Theorem 1, D ) is generated by the weight enumerators x) of d + of n lengths, 16,, +3 = 1 By calculating the dimension of the homogeneous part for each n, we get the result This completes the proof Theorem 4 The ring is the normalization of D ) in its field of fractions Proof The ring is generated by the s and W d + g ) 4 Because of dim = dim D) = n ) 4), ) d W + should be written as a linear combination of the s We can say more ) 7 ) The product W is indeed in D ) 0 by calculation At any rate, we see that W) and D ) have the same field of fractions Since it can be shown that is a root of a monic quadratic equation over D ) by explicit calculation, is integral over D ) We give the mentioned forms above explicitly in Appendix Since the invariant ring of a finite group is normal, so is This completes the proof We conclude this paper with some comments As a finite analogue of Eisenstein series, we studied E-polynomials cf [, 9]) Since is a unique Type II code of length, we obtain the identity between an E-polynomial of weight and The resulting identity seems to be non-trivial Let τ be an element of the Siegel upper-half space of genus g For α, F g, we define a Thetanullwert [ α θ τ) = ] { 1 exp π 1 t p + 1 ) α τ p Z g p + 1 α ) + t p + 1 α ) 1 } 7

8 [ ] α We put f α τ) = θ τ) It is known cf [4]) that 0 θ [ ] ) γ τ) = 1) α f α+γ τ)f α τ) α F g Under the theta map x α f α τ), we derive the theta series ϑ g) τ) of an even unimodular D n + lattice D n + from the weight enumerator of in genus g Therefore we have that ϑ g) D + n τ) = f α τ) : α F g ) = 1 g 1) α f α+γ τ)f α τ),γ F g = 1 g,γ F g θ α F g [ γ which was given in [5] without coding theory ] τ) ) n n/ Appendix: Expressions of We shall denote by d n instead of and by C instead of C For example, d7 means ) 7 In the first formula, if we divide both sides by d 7, we get a rational expression of by the d n s In the second formula, we can see that is a root of a monic quadratic equation over D ) d 7 = / d / d /5530 d 3 d /69136 d 4 d /41416 d 4d /0740 d d /1370 d d 3 d / d d 4 d /33334 d d /135 d d / d d /5150 d d 4 d /137 d 3 d /963 d 3 d 4 d /515 d 4 d /64 d 4 d / d 5 d /33150 d 6 d / d 7 d / d 10

9 g 4 = 53361/97 d /464 d /14640 d d / d d /19944 d 3 d / d /15 d 4 39/1464 d 3 ) References [1] Broué, M, Enguehard, M, Polynômes des poids de certains codes et fonctions thêta de certains réseaux, Ann Sci École Norm Sup 4) 5 197), [] Duke, W, On codes and Siegel modular forms, Internat Math Res Notices 1993, no 5, [3] Gleason, AM, Weight polynomials of self-dual codes and the MacWilliams identities, Actes du Congrès International des Mathématiciens Nice, 1970), Tome 3, pp Gauthier-Villars, Paris, 1971 [4] Igusa, J, On the graded ring of theta-constants, Amer J Math ), [5] Igusa, J, Schottky s invariant and quadratic forms, E B Christoffel Aachen/Monschau, 1979), pp 35-36, Birkhäuser, Basel-Boston, Mass, 191 [6] Kneser, M, Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen, Math Ann ), [7] Nebe, G, Rains, E, Sloane, NJA, Self-dual codes and invariant theory, Algorithms and Computation in Mathematics, 17 Springer-Verlag, Berlin, 006 [] Oura, M, Eisenstein polynomials associated to binary codes, Int J Number Theory 5 009), no 4, [9] Oura, M, Eisenstein polynomials associated to binary codes II), Kochi J Math ), [10] Runge, B, On Siegel modular forms I, J Reine Angew Math ), 57-5 [11] Runge, B, On Siegel modular forms II, Nagoya Math J ), [1] Runge, B, Codes and Siegel modular forms, Discrete Math ), no 1-3, [13] Shephard, GC, Todd, JA, Finite unitary reflection groups, Canadian J Math ), [14] Witt, E, Eine Identität zwischen Modulformen zweiten Grades, Abh Math Sem Hansischen Univ 14, 1941), Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa , Japan address: makotofkanazawa@gmailcom address: oura@sekanazawa-uacjp 9