On the density of cyclotomic lattices constructed from codes
Keywords : Lattice sphere packings, Minkowski-Hlawka bound, cyclotomic fields, linear codes.
Mathematics Subject Classification : 11H31, 11H71.
This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the "Investments for the future" Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02)
Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor for infinitely many dimensions . Here we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A.
The sphere packing problem in Euclidean spaces asks for the biggest proportion of space that can be filled by a collection of balls with disjoint interiors having the same radius. Here we focus on lattice sphere packings, where the centers of the balls are located at the points of a lattice, and we denote by the supremum of the density that can be achieved by such a packing in dimension . Let us recall that the exact value of is known only for dimensions up to [CSB87] and for dimension ([CK09]). For other dimensions, only lower and upper bounds are known. Moreover, asymptotically, the ratio between these bounds is exponential.
Here we focus on lower bounds. The first important result goes back to the celebrated Minkowski-Hlawka theorem [Hla43], stating the inequality for all , where denotes the Riemann zeta function. Later, Rogers [Rog47] improved this bound by a linear factor: he showed that for every , with . The constant was successively improved by Davenport and Rogers [DR47] (), Ball [Bal92] () and Vance [Van11] ( when is divisible by ). Recently Venkatesh has obtained a more dramatic improvement [Ven13], showing that for big enough, . Most importantly, he proves that for infinitely many dimensions , , thus improving for the first time upon the linear growth of the numerator.
Unfortunately, all these results are of existential nature: their proofs are non constructive by essence, due to the fact that they generally use random arguments over infinite families of lattices. It is then natural to ask for effective versions of these results. It is worth to explain what we mean here by effectiveness. Indeed, designing a practical algorithm, i.e running in polynomial time in the dimension, to construct dense lattices appears to be out of reach to date. More modestly, one aims at exhibiting finite and explicit sets of lattices, possibly of exponential size, in which one is guaranteed to find a dense lattice.
In this direction, the first to give an effective proof of Minkowski-Hlawka theorem was Rush [Rus89]. Later, Gaborit and Zémor [GZ07] provided an effective analogue of Roger’s bound for the dimensions of the form with a big enough prime number. In both constructions, the lattices are lifted from codes over a finite field, and run in sets of size of the form , with a constant.
Let us now explain with more details two ingredients that play a crucial role in the proofs of the results above. The first one is Siegel’s mean value theorem [Sie45] which in particular states that, on average over the set of -dimensional lattices of volume ,
It follows that, if , then there exists a lattice such that , i.e such that the minimum norm of its non zero vectors is greater than . The density of the sphere packing associated to then satisfies
It is worth to point out that the same reasoning holds if , because lattice vectors of given norm come by pairs . From this simple remark we get
which is essentially Minkowski-Hlawka bound.
The second idea follows almost immediately from the previous observation: considering lattices affording a group of symmetries larger than the trivial should allow to replace the factor in the numerator by a greater value. To this end, one needs a family of lattices, invariant under the action of a group, for which an analogue of Siegel’s mean value theorem holds. This idea is exploited in [GZ07], [Van11] and [Ven13]. In particular, this is how Venkatesh obtains the extra term, by considering cyclotomic lattices, i.e lattices with an additional structure of -modules. It turns out that, for a suitable choice of , one can find such lattices in dimension .
In this paper, we consider cyclotomic lattices constructed from codes, in order to deal with finite families of lattices. To be more precise, the codes we take are the preimages through the standard surjection associated to a prime ideal of
of all one dimensional subspaces over the residue field .
Our approach is simpler and more straightforward than the previous ones in several respects. On one hand, the analogue of Siegel’s mean value theorem in our situation boils down to a simple counting argument on finite sets (see Lemma 4). On the other hand, the group action, which is, as in [GZ07], that of a cyclic group, is in our case easier to deal with, because it is a free action. As a consequence, we can cope with arbitrary orders , while Gaborit and Zémor only consider prime orders.
Our main theorem is an effective version of Venkatesh’s result:
For infinitely many dimensions , a lattice such that its density satisfies
can be constructed with binary operations.
This result follows from a more general analysis of the density on average of the elements in the families of -cyclotomic lattices described above, see Theorem 2 and Proposition 1 for precise statements.
A lattice is said to be symplectic if there exists an isometry exchanging and its dual lattice, and such that . Symplectic lattices are closely related to principally polarized Abelian varieties. In [Aut15], Autissier has adapted Venkatesh’s approach to prove the existence of symplectic lattices with the same density. We show that, with some slight modifications, our construction leads to symplectic lattices, thus providing an effective version of Autissier’s result (see Theorem 3 and Corollary 2 ).
The article is organized as follows: Section 2 recalls basics notions about lattices and cyclotomic fields, and introduces the construction of cyclotomic lattices from codes. In Section 3 we state and prove the main results discussed above. Section 4 is dedicated to the case of symplectic lattices.
I am most grateful to Christine Bachoc for introducing me to this problem, and for her support all along this work. I would also like to thank Arnaud Pêcher and Gilles Zémor for fruitful discussions, and Pascal Autissier for useful remarks that lead to improvements on the first version of the paper.
2. Notations and preliminaries
2.1. Lattices in Euclidean spaces
Let be a Euclidean space equipped with the scalar product . We denote by the norm associated to this scalar product, by the dimension of , and by the closed ball of radius in E:
By Stirling formula, we have
where means . Thus, if , we get that
A lattice is a free discrete -module of rank (for a general reference on lattices, see e.g [CSB87]). A fundamental region of is a region such that for any , the measure of is , and . The volume of is defined as the volume of any of its fundamental region. The Voronoi region of is the particular fundamental region:
We denote by the minimum of :
and by its covering radius:
Taking balls of radius centered at the points of , we get a packing in , i.e a set of spheres with pairwise disjoint interiors. The density of this packing is given by
Finally, we define , the dual lattice of the lattice :
2.2. Cyclotomic fields
Let be the cyclotomic field , where is a primitive -th root of unity. This is a totally imaginary field of degree over . Let us define . The trace form where denotes the trace form of the number field induces a scalar product on , denoted by , giving the structure of a Euclidean space of dimension . We refer to [Was97] for general properties of cyclotomic fields.
For every fractional ideal , we will use the same notation for the lattice in which is the image of under the natural embedding . We will need informations about lattices defined by fractional ideals of .
The volume of is by definition the square root of the absolute value of the discriminant of . It is well known (e.g [Was97]) that for the cyclotomic fields
where is the set of prime numbers.
It is easy to see that the minimum of is : indeed and the arithmetic geometric inequality gives for all . For the minimum and the covering radius of general fractional ideals, we will apply the following estimates:
Lemma 1 ([Flu06], propositions 4.1 and 4.2.).
Let be a fractional ideal of , where is a number field of degree over . Then we have :
2.3. Cyclotomic lattices constructed from codes
A standard construction of lattices lifts codes over to sublattices of , this is the well known Construction A (see [CSB87, Chapter 7]). Here we will deal with a slightly more general construction in the context of cyclotomic fields.
Let us consider as before and the Euclidean space associated with . Let be a prime ideal of lying over a prime number which does not divide . Then the quotient is a finite field of cardinality .
Let . We still denote by the scalar product induced on the -dimensional -vector space by that of . Let be a lattice in which is a -submodule of . We consider the canonical surjection
The norm on associated with induces a weight on the quotient space : if ,
The quotient is a vector space of dimension over the finite field . We will call a -subspace of a code. We denote by its dimension and by its minimal weight, with respect to the weight defined above. Finally we denote by the lattice obtained from
and give in the following lemma a summary of its properties:
Let be a code of of dimension and minimal weight . Then :
The volume of is
The minimum of is .
If , the packing density of is:
where is the dimension of .
The lattice contains the lattice and we have:
and (iii) follow directly from the definitions.
To conclude this subsection, we state a lemma that relates the Euclidean ball and the discrete ball .
Assuming , we have:
If , then
Let such that . We want to prove that has exactly one representative which satisfies . Indeed, if with and , we have with . Then , a contradiction.
Let us consider
where is the Voronoi region of . The volume of is
so the wanted inequalities will follow from the inclusions
Let us start with the second inclusion. If , by definition of the covering radius, we have
so if , . For the first inclusion, let be such that . If denotes the closest point to in , we have and , so that .
It follows directly from Lemma 2.
3. The density of cyclotomic lattices constructed from codes
In this section, we introduce a certain family of lattices obtained from codes as described in the previous subsection, and show that for high dimensions, this family contains lattices having good density.
As before, , . Let us set and consider the Euclidean space , of dimension , in which we fix .
We denote by the set of the -lines of , and by the set of lattices of constructed from the codes in :
The following lemma evaluates the average of the value of over the family :
It is a straightforward computation:
There is exactly one line passing through every non zero vector in . So
From now on, will vary with , so we adopt the notation instead of . We show that the family of lattices contains, when is big enough and when grows in a suitable way with , lattices having high density.
For every , if , then for big enough, the family of lattices contains a lattice satisfying
We start with a technical lemma.
Let . If , then
For big enough, .
Now we can prove Theorem 2.
Proof of Theorem 2.
Now applying (i) of Lemma 5, we have , and so, for big enough,
Now comes the crucial argument involving the action of the -roots of unity. From (4), there is at least one code in which satisfies . Because the codes we consider are stable under the action of the -roots of unity, which preserves the weight of the codewords, and because the length of every non zero orbit under this action is , we can conclude that , and so by (iii) of Lemma 3 that,
Theorem 2 shows that for every big enough dimension of the form our construction provides lattices having density approaching , thus larger than with . A particular sequence of dimensions leads to a better lower bound:
For infinitely many dimensions, the family contains a lattice satisfying
To get the optimal gain between and , we take , where is a positive real number, which tends to infinity. Thanks to Mertens’ theorem [Har], we can evaluate:
where is the Euler-Mascheroni constant which satisfies .
So we get
Finally we evaluate the complexity of constructing a lattice with the desired density:
Let . For every , the construction of a lattice satisfying
requires binary operations.
We need to find a prime ideal such that satisfies the condition required in . Let us recall that where is the prime number lying under , and is the order of in the group (see [Was97]). We will restrict our attention to the case , i.e when . In that case, decomposes totally in , and . We use Siegel-Walfisz theorem in order to give an upper bound for the smallest such prime number:
For big enough, there is a prime number congruent to such that:
Let us denote by the number of primes such that . Siegel-Walfisz theorem (see [IH04]) gives that for any :
where the implied constant depends only on , and . Applying this theorem to , , and we get
We have , which grows faster than the error term since , and thus ensures the existence of a prime between and . ∎
Proof of Proposition 1.
The corresponding family of lattices has elements. By construction, each of these lattices is generated by vectors with coefficients which are polynomial in . So, the cost of computing their density, which can be done with operations, following [HPS], is negligible compared with the enumeration of the family. ∎
4. Symplectic cyclotomic lattices
For a survey about symplectic lattices, we refer to [Ber97]. Here we briefly introduce this notion.
Let be a Euclidean space, and a lattice in . Then an isoduality is an isometry of such that . If affords an isoduality, then it is called isodual. If moreover satisfies , then is called symplectic.
Now we explain how to change the lattice in such a way that our construction provides symplectic lattices.
where . The volume of is now
Let us define the map
It is clear that is an isometry, and that .
In the following lemma, we show that the lattices we defined in Definition 1 are now symplectic:
If is a -line of , then the lattice is symplectic.
Let us prove that . Let us take . We have to show that for every , , that is
According to the definition of , we have with and . So there exists such that
This implies that
so that (8) is satisfied.
To conclude the proof it is enough to notice that , which implies . ∎
We again consider the set of lines of . It is clear that the result of Lemma 4 remains valid for this new family of codes. The general strategy underlying the proof of Theorem 2 applies to the family of lattices associated to these codes, so that we get analogues in this context :
For every , if , then for big enough, the family of symplectic lattices contains a lattice satisfying
For infinitely many dimensions, the family contains a symplectic lattice satisfying
If , then
For big enough, .
Let us set and . Then and the covering radius of is . So we have to bound both covering radii and . Applying (ii) of Lemma 1, and because , we have, for ,
which tends to when goes to infinity, by hypothesis.
Let us set and . Then , and clearly . Then, applying (i) of Lemma 1, since , we have, for ,
The hypothesis on ensures in particular that for big enough, satisfies , and thus
As the condition on the growth of does not change, the estimation for the complexity of construction in this context is the same:
Let . For every , the construction of a symplectic lattice satisfying
requires binary operations.
- [Aut15] Pascal Autissier. Variétés abéliennes et théorème de Minkowski-Hlawka. Manuscripta Mathematica, 2015.
- [Bal92] Keith Ball. A lower bound for the optimal density of lattice packings. Internat. Math. Res. Notices, (10):217–221, 1992.
- [Ber97] Anne-Marie Bergé. Symplectic lattices. Contemporary Math., 272:9–22, 1997.
- [CK09] Henry Cohn and Abhinav Kumar. Optimality and uniqueness of the Leech lattice among lattices. Ann. of Math. (2), 170(3):1003–1050, 2009.
- [CSB87] J. H. Conway, N. J. A. Sloane, and E. Bannai. Sphere-packings, Lattices, and Groups. Springer-Verlag New York, Inc., New York, NY, USA, 1987.
- [DR47] H. Davenport and C. A. Rogers. Hlawka’s theorem in the geometry of numbers. Duke Math. J., 14:367–375, 1947.
- [Flu06] Eva Bayer Fluckiger. Upper bounds for euclidean minima of algebraic number fields. Journal of Number Theory, 121(2):305 – 323, 2006.
- [GZ07] Philippe Gaborit and Gilles Zémor. On the construction of dense lattices with a given automorphisms group. Ann. Inst. Fourier (Grenoble), 57(4):1051–1062, 2007.
- [Har] G. H. Hardy. Note on a Theorem of Mertens. J. London Math. Soc., S1-2(2):70.
- [Hla43] Edmund Hlawka. Zur Geometrie der Zahlen. Math. Z., 49:285–312, 1943.
- [HPS] Guillaume Hanrot, Xavier Pujol, and Damien Stehlé. Algorithms for the shortest and closest lattice vector problems. In In Yeow Meng Chee, Zhenbo Guo, San Ling, Fengjing Shao, Yuansheng Tang, Huaxiong Wang, and Chaoping Xing, editors, IWCC, volume 6639 of Lecture Notes in Computer Science, pages 159–190. Springer.
- [IH04] Kowalski E. Iwaniec H. Analytic number theory. 2004.
- [Rog47] C. A. Rogers. Existence theorems in the geometry of numbers. Ann. of Math. (2), 48:994–1002, 1947.
- [Rus89] J. A. Rush. A lower bound on packing density. Invent. Math., 98(3):499–509, 1989.
- [Sie45] Carl Ludwig Siegel. A mean value theorem in geometry of numbers. Ann. of Math. (2), 46:340–347, 1945.
- [Van11] Stephanie Vance. Improved sphere packing lower bounds from Hurwitz lattices. Adv. Math., 227(5):2144–2156, 2011.
- [Ven13] Akshay Venkatesh. A note on sphere packings in high dimension. Int. Math. Res. Not. IMRN, (7):1628–1642, 2013.
- [Was97] Lawrence C. Washington. Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.