Polynomial-time construction of spherical codes

polynomial-time construction of spherical codes Jacques Stern t

Gilles Lachaud*

Abstract

We give a simple lower bound for the dimensions of the families of polynomially constructible spherical codes of given minimal angle ~o, deduced from the analog of the Katsman-Tsfasman-Vl~du!i bound for linear codes. In particulax the supremum rpol of the numbers log2 CardX/dim X, where X ranges over all polynomially constructible families of spherical codes with ~ _> Ir/3, is such th~.t rpot > 2/15.

1

spherical

codes

We note x.y the usual scalar product of two vectors x and y in the euclidean space R n, thus II x 112= x.x is the square of the length of x. A spherical code is a set X of points lying on the unit sphere

s "-1 = {~ ~ l~" [ II • II== 1}. We refer to [2] and to the references therein for the results on spherical codes. If x and y are in S n - 1 we set

~(., y) = arccos..y,

(0 R, i'--*OO

)im p(Xi) > p. $--'~ OO

We set

R(p) = sup{R I (p, R) e u}. It is known from the results of Rankin (cf. [2], p. 27) that R(p) = 0 if 2 < p < 4. Now let p < 2; Chabanty in 1953 (cf.[1]) and Shannon in 1959 (cfl [7]) have given a lower bound for R(p), namely R(p) > Rcs(p), with 1

22cs(P) = log2 sin ~o(p)" On the other hand, the best upper bound known up to now is the Kabatyansky-

Levenshtein bound (eft [3]): R(~) < 1 + sin__________log ~ 2 1 + sin ~ 2 sin ~o 2 sin ~o -

1 - sin ~ log2 1 - sin 2sin ~o 2 sin ~o

Nevertheless, the spherical codes used to get these bounds are constructed by an exhaustion process, hence the complexity of construction is doubly exponential. We are concerned here by spherical codes with polynomial complexity of construction. Here is a precise definition. Let ~ be a finite alphabet. We consider spherical codes X C S m-1 which are images of maps F : ~k _+ Srn-1. We say that a family (X/)i>0 of spherical codes is polynomially consl~ctible if there is a sequence (Fi)i>_.0 of maps ~i : ]~kl _.~ s m i - I

such that: 1. The map Fi is one-to-one from Ek~ to X~; 2. For every a E ~k,, the point Fi(a) is computable from i and a in polynomial time with respect to the dimension rni of Xi.

220

2

the concatenation

b o u n d for s p h e r i c a l c o d e s

We define now the domain ofpolynomially constructible spherical codes as the set Upol of couples (p, R) E [0, 2] × [0, +¢o[ such that there exists a potynomiatly constructible family (Xl)i>_0 of such codes satisfying (1). We set Rpo~(p) = S u p { R l (p, R) e Upon}; this is the asymptotic bound for polynomially constructible spherical codes. Let q be a number which is a power of a prime number p, and consider a block code C with parameters [n,k,d] over Fq ; this means that C is a subset of F~, that k = log 2 CardC, and that the minimal Hamming distance between two elements of C is equal to d. On the other hand, take a spherical code X which is the image of a one-to-one map F : Fq -+ S m-1. The spherical code obtained by concatenation of X (the inner code) and (:3 (the outer code) is the code Y

C S n r n - 1 C p nrn :

l:lrn X ... X R m

whose elements are the vectors y(a) = ~ n (F(al),..., F(an)),

i f a = (at, ..., an) E C.

This is the simplest case of the concatenation method of [9]. The parameters of the code Y are dim Y = n m ,

C a r d ( Y ) = q k,

p(Y)> dp(x),

P~(Y)--kp~(x).

If we apply this construction by taking as outer codes the modular Goppa codes constructed by Katsman, Tsfasman, Vl~du~, and Zink (cf. [4], [8]), we get: T h e o r e m 1 If Q is an even power of a prime number, if X is a spherical code wi~h Q elements, and if p >_ O, let

F(x,p) = n(x)(1

p

p(X)

1

1)

and define /~, (p) = max F(X, p), where the maximum is taken over all spherical codes X whose number of elements is an even power of a prime number. Then

>_n,(p).

221

R e m a r k . The preceding result can be generalized. Let there be given an increasing family X of spherical codes in S N-1 : Xl C ... C X~ and denote by Mi the number of elements of the code Xl. Assume Mi = Qt ... Qi, where Qi is an even power of a prime number, and let

F(X,P) = f i ~o~iiF(Xi,P); i=1

then

e o (p) > F(X,p).

3

explicit concatenation b o u n d s

The value p = 1 is of particular interest. For any number n, the kissing humbert(n) is the maximum number of elements of a spherical code in P~" with minimum angle greater or equal to ~r/3 ; it is also the greatest number of an arrangement of spheres, that is a set of spheres of equal radius that touch one sphere in the n-dimensional space (cf. [2], p. 21). We set r = R(1) = limsup l°g2 r(n) i--*oo

n

We know, from the Kabatyansky-Levenshtein bound (cf. [3]) that r _< 0.401414, and the Chabauty-Shannon "nonconstructive " bound gives

r >_Rcs(1) = 1 -

log 2 3 = 0.2075...

Denote now by vx(n) the number of points of a polynomially constructible family of spherical codes (Xi)i>>_owith minimum angle r / 3 . The result of Leech (cf.[2], p. 24) asserts that such a family can be found with

1

log~ rx(n) > ~(log 2 n)(log2(n ÷ I)). The asymptotic polynomially construc~ible kissing number rpot = Rpoz(1) is the supremum of the numbers log 2 rx (n) n

when X ranges over all polynomially constructible families of spherical codes. The best previously known result was rpot > 0.003..., obtained from the Weldon and Sugiyama et

222

at. codes (cf.[2], p. 27). Now if in theorem I we take as inner code the (16,28,3/2) sphericM code X constructed from the Nordstrom-Robinson binary code .M16 with parameters [16, 8, 6], then we get 1.14 2 F ( X , p ) = ~(~-~ - ~p),

2 F ( X , 1) = ~-ff = 0.133...,

and consequently : T h e o r e m 2 The asymptotic polynomially construciible kissin9 number is such that 2

rpo, >_ "i-5" If we apply repeatedly theorem 1, considering various spherical codes depending on the value of p, then : T h e o r e m 3 One has 1 R cs(;) n,(p) >_ ~in other words

1

n4p(~o)) > ~log, We have also

i f p < 1.535...; 1

i f ~ ~ 76034 '.

1 1 R,(p(~o)) > ~]og2~n~

0.0034.

The last assertion has to be compared with the upper bound of Kabatyansky-Levenstein (cf.[3]) : 1

1

0.0099.

Now we examine the asymptotic behaviour at the ends. The Chabauty-Shannon bound satisfies 1 I

Rcs(p) ~ ~log~ ~

ifp ~ o,

Rcs(2 - x) ,-~ 8 ll~-~x2

i f z -+ O,

but recall that that bound holds for nonconstructive families of spherical codes. As long as we pay attention to the asymptotic bound for polynomially constructible families of spherical codes, we have : T h e o r e m 4 If p ---* O, then 1

R,,(p) >_ O l o g u ~ + o(1); if z --~ O, then

~ ( 2 - ~) _> ~ 3 iog~ !~ + o(1).

223

The first assertion is proved by taking as inner codes in theorem 1 those obtained from the Es lattice, and the second by taking in the same way the projective Reed-Muller codes. The detailed proofs of the proceding results will appear in [5] and [6].

References [1] Chabauty~ C., R&ultafs sur l'empilement de calories dgales sur use pgrisph~re de R " el correction fiun travail antgrieur, C.R.A.S. 236 (1953), 1462-1464. [2] Conway, J.H., Sloane, N.J.A., Sphere Packings, Lattices and Groups, Grund. der math. Wiss. 290, Springer, New-York-Heidelberg, 1988. [3] Kabatyansky, G.A., Levensteha V.I., Bounds for packings on a sphere and in space, Problemy Peredachi Informatsii 14 (1979), 3-25 ; = Problems of Information Transmission 14 (1979), 1-17. [4] Katsman, G.L., Tsfasman, M.A., Vl~duL S.C., Modular Curves and Codes with polynomial complexity of construction, Problemy Peredachi Informatsii 20 (1984), 4755 ; = Problems of Information Transmission 20 (1984), 35-42. [5] Laehaud, G , Stern, J., polynomial-time construction of codes f : linear codes with almost equal weights, preprint. [6] Laehaud, G., Stern, J., polynomial-time construction of codes H : spherical codes and the kissing number of spheres, preprint. [7] Shannon, C., Probability of error for optimal codes in a gaussian channel, Bell System Technical Journal 38 (1959), 611-656. [8] Tsfasman, M.A., Vl~du~, S.G., Algebraic-Geometric codes, Kluwer Acad. Pub., Dordrecht, 1991. [9] Zinoviev, V.A., Litsyn, S . N , Portnoi, S.L., Concatenated codes in euclidean space, Problemy Peredachi Informatsii 25 (1989), 62-75 ; = Problems of Into. Transmission 25 (1989), 219-228.