On the nonexistence of periodic tilings with cubistic cross-polytopes

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On the nonexistence of periodic tilings with cubistic

cross-polytopes

Citation for published version (APA):

Post, K. A. (1979). On the nonexistence of periodic tilings with cubistic cross-polytopes. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 79-WSK-07). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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ONDERAFDELING DER WISKUNDE DEPARTMENT OF MATHEMATICS

On the nonexistence of periodic tilings with cubistic cross-polytopes

by

K.A. Post

T.H. - Report 79-WSK-07 November 1979

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1. In troduction

with cubistic cross-polytopes by

K.A. Post

The existence of periodic tilings of n-space (n ~ 2) with congruent cubistic analogues of the cross-polytopes of radius e (e ~ 1) has been investigated by several authors:

GOLOMB and WELCH ([6J) gave examples for the case n :::: 2 (e arbitrary) and for the case e = 1 (n arbitrary). They also proved the nonexistence for the parameter values (n,e) (3,2) and n ~ 3, e ~ p for some function p of n,

n n

the numerical values of which were not known. They conjectured that P n 2 for all n ~ 3.

Explicit bounds were found by POST ([9J, [10J) who showed among other things, that P

n 2 (3 ::; n ::; 6) P7::; 3, Pn S Jm/2 - ~12

-

~ (n ~ 8). Cf. also MULLER

([8J).

Further research was done by ASTOLA ([lJ, [2J), BASSALYGO ([3J) and LENSTRA ([7J), but the results obtained by these authors all concern tilings with periods having a specific kind of prime decomposition.

In this paper we shall show among other things that p

s

3 (8

s

n ::; 10) and n

that p

In

~ 0,3735 as n + ~

n

2. Basic concepts. Generating functions

The cubistic cross-polytope of radius e ~ 0, inE I with center

n

(x 1' ... ,x) E ~ is defined to be the union of all unit cubes centered in

n n

those points (y 1 I • • • ,y ) E ~ tha t satisfy the inequali ty

n n

n

L

I

xi - Y i

I

s

e . i==l

(t 0 , ••• ,n-1) (see Fig. 2) •

j=O

Combination of i) and ii) indicates that i t is important to know how many Hamming spheres of various sizes can be combined to form a decomposition

n of {O,l} .

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We conclude this section by giving the generating function for the number of vertex points of various types in one orthant of a given cubistic cross-polytope (see [9]).

. \'t (n.). th

Let gn,e,t denote the number of vertex po~nts of type Lj=O J ~n one

Or

ant of a cubistic cross-polytope of radius e in n-space. Then

00

I

e=O g z n,e, t e z t n (1 - z)

3. Optimal packing of Hamming spheres in {0,1}8

Hamming spheres in binary 8-space have the following volumes radius volume

o

1 1 9 2 37 3 93 4 163 5 219 6 247 7 255

A packin9 IT of {O,l}8 is defined to be a partition of {0,1}8 into diSjoint Hamming spheres.

A packing IT, consisting of m. spheres of radius i (i

~ O, .•• ,7) say, is

call-ed optimal if no packing exists with n. spheres of radius i, satisfying

~

nO < m_O' n₁> m

1 and ni = mi (i F 0 11). We shall investigate systematiaally, wht Ch combinations (mi)l=o admit optimal packings.

I . 25S .1 (i.e. m7 == 1, mO centers.

I, m. == 0 (i

F

0,7}) trivial, take antipodal

~

II. 247.9 (m

6 == I, m1 I, m. ~ == 0 (i

F

1,6» trivial, antipodal centers.

111«. 219.37 trivial, antipodal centers. 28

III@.

219.9.1 (m

S == 1, m1 1, mO == 28, mi == 0 (i

F

0,1,5». Assume that (1 1 1 1 1 1 1 1) is the radius-5-center. All radius-1-centers must have weight ~ 1, hence cannot have mutual distance ~ 3. So m

1 ~ 1. Obviously, a packing with the given m.-values exists.

1.

IV«. 163.93. Trivial, antipodal centers. 56

1V~. 163.37.1 • Let (1 1 1 1 1 1 1 1) be the radius-4-center. Radius-2-centers must have weight <; 1 (so there is at most one radius-2-center), and w.l.o.g. we have to distinguish between the following two cases

iJ ii) (r 4) (1 1 1 1 1 1 1 1) (r 2) (1 0 0 0 0 0 0 0) (r '" 4) (1 1 1 1 1 1 1 1) (r == 2) (0 0 0 0 0 0 0 0)

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In case i) let an additional radius-1-center have weight a in the first coor-dinate, weight b in the remaining 7 coordinates. Then we get the inequalities

(1 - a) + (7 - b) 2! 6}

, so 9 - 2a 2! 10, a contradiction • (1 - a) + b ~ 4

In case ii let a radius-1-center have weight a. The resulting inequalities

8 -

a 2!

6}

yield 8 2! 10, which is impossible • a 2! 4

So there cannot be an additional radius-1-sphere.

IVy. 163.94.157. Assuming that (1 1 1 1 1 1 1 1) is the radius-4-center we must locate the radius-1-centers in points of weight ~ 2. In order to keep their distances ~ 3 these points must have disjoint supports, and at most one of them can have weight 1. So there can be 4 radius-1-spheres and no more.

2 70

Va. 93 .1 • Assume that (1 1 1 1 1 1 1 1) is radius-3-center. The other radius-3-center must have weight 1 or 0, and we apply the same method as we did in IV •

Case i) yields for an additional radius-l-center (1 - a) + (7 - b) 5

(1 - a) + b ?: 5} so 9 - 2a ;::: 10, a contradiction

Case ii) implies 8 - a ?: 5

it :: 5}' hence B 2 10, a contradiction •

VS. 93.37.97.163. Assume that (1 1 1 1 1 1 1 1) is the radius-3-center. Ra-dius-2-centers then must have weight ~ 2. so only one of them can be located, and without loss of generality there are three cases to be considered. We still use the same kind of argumentation as we did in IV •

Case i) (r 3) (1 1 1 1 1 1 1 1 ) (r 2) (1 1

a o

0

a a

0) (r 1) a b (weights) Now (2 - a) + (6 - b) 2 5 4} so 10 - 2a 2 9

,

(2 - a) + b '>

and the solutions are (a,b) == (0,2) , (0,3) • Because A(6,4,2) + A(6 ,4, 3)

₌

3 + 4

=

7 (cL [5 J) we see that at most 7 radius-1-spheres are possible in this case.

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Case i i (r 2) (1 I 1 1 I 1 1 1) (r 2) (1 0

o

0

a a

0 0) (r 1 ) a b (weights)

.

We find (1 - a) + (7 - b) ~

:}

(1 - a) + b ~

wi th one sinqle solutiun (a,b) = (0,3).

Accordinq to [5J, A(7,4,3)

=

7, and the incidence matrix of the Fano-plane locates the 7 radius-i-centers.

0 1 0 0 1 1)

Obviously, only one radius-1-center can have weight 0 or weight 1. However, such a center would not allow a maximal weight-2-set. So the packing is appa-rently optimal.

VIa. 374.1108. nlree radius-2-centers in {a,l}8 always exhibit the distance

pattern 5,5,6, but allow a fourth radius-2-center uniquely. W.l.o.g. we may locate the centers

(r 2) (1 1 1 1 1 1 1 1)

(r = 2) (1 1 0

a a

0

a

0)

(r

₌

2) (0 0 1 1 1 0 0 0) (r 2) (0 0 0

a

0 1 1 1)

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Let (r = 1) a b c (weights). This yields the inequalities

(2 - a) + (3 - b) + (3 - c) ~ 4 (2 - a) + b + a + (3 - b) + c ~ 4 c ~ 4 a + b + (3 - c) ~ 4

From the last three inequalities i t follows that c ~ 2, a ~ 1, b ~ 2, a con-tradiction with the first inequality.

VIS. 373.99.164. Without loss of generality we may assume (see VIa) the ra-dius-2-centers to be located as follows:

let Hence and we Class (r 2) (1 1 1 1 1 (r ::; 2) (1 1 0 0 0 (r 2) (0 0 1 1 1 (r = 1) a b 1 1 1)

o

0 0)

o

0 0), c (2 - a) + (3 -bl ₊ (3 - c) ~ (2 - a) + b + a + (3 - b) ₊

get the solutions a b c 1 0 1 1 2 2 t t (1) (2) 0 0 1 0 0 0 2 3 3 t ( 3) c ~ c ~ 0 1 3 (weiqhts) 4 4 4

Since all points in clas:3 (3) have relative distances ::;; 2 only one radius-l-center can be chosen in class (3).

Class (2) can contain at moEt 3 radius-1-centers that w.l.o.g. can be located in the points

(O 0 '1 0 0 0 1 1)

(2) (0 0 0 1 0 1 0 1)

(0 0 0 0 1 1 1 0) •

In the same way, class (1.) can contain at most 2

*

3 centers, that however can be chosen compatible with the choice made above in an essentially unique way

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( 1 0 1

o

0 1 0 1) (1') (1 0

o

1 0 1 1 0) (1 0 0 0 1 0 1 1) (0 1 1 0 0 1 1 0) (1") (0 1

o

1 0 0 1 1) (0 1

o

0 1 1 0 1)

Since this configuration is unique and does not leave place for a center in class (3) the optimality of the given packing is proved.

3 2 914 156 h "d f th radius-2-centers Vly. 7 . • • Now we ave to cons~ er our cases: e

can have distance 5,6,7 or 8.

(r 2) (1 1 1 1 1 1 1 1)

(r == 2) (1 1 1 0 0 0 0 0) let (r

=

1) a b The resulting inequalities

(3 - a) + (5 - b) ;;:: 4

(3 - a) + b ;;:: 4

have the solutions

a 0 0 0 0 1 1 b 1 2 3 4 2 3 C 2 4 1 3 0 1 0 C 3 0 3 I 4 0 1

In this diagram the entries C

2 and C3

(weights) •

denote the number of points in {0/l}8 that have weight 0 in the first 3 coordinates, weight 2 resp. 3 in the last S coordinates, that are covered by radius-1-spheres centered in the points given by (a,b).

Let Po be the number of radius-I-centers chosen with a = 0, and P

1 the cor-responding number with a = 1. Addition of the C

2 and the C3 values yields

On the other hand, since a maximal set of weight-2 and weight-3 vectors in {O,l}S that have minimum distance;;:: 3, contains 4 elements, we see in addi-tion that

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Case ii) let Now the and have Class (r 2) (1 1 1 (r 2) (1 1 0 (r 1 ) a inequalities are (2 - a) + (6 - b) (2 - a) + b the solutions a b 1 0 3 3 1 1 (1)

o

0 2 4 4 4 (2) 1 1 1 1 1) 0 () _{0 0 0)} b (weights) ~ 4 ~ 4 Here C

3 denotes the number of points having weight 0 in the first 2 coordi-nates, weight 3 in the remaining coordicoordi-nates, that are covered by a radius-I-sphere centered in a point of given (a,b) weight.

Let P. be the number of centers chosen in class i) (i

~

vectors on the last 6 coordinates yields

1,2). Counting weight-3

On the other hand, since A(6,4,3) 4 (cf. [5]) we obtain P 1 :s; 2

*

4 + 1

*

4 12, so that Pl + P 2 :s; 14. Case iii) (r 2) (1 1 1 1 1 1 1 1) (r 2) (1 0 0 0 0 0 0 0) let (r 1) a b

The corresponding inequalities are (1 - a) + (7 - b) ~ 4

(1 - a) + b ~ 4

(weights) •

and have the solutions (a,b) (0,3), (0,4). Let a set of radius-l-centers be chosen. Replace the first coordinate of every center by a parity bit. Then we get a set of weight-4-vectors, distance

~

4. in {0.1}8. Since A(8,4,4)

=

14 (cf. [5J) the result follows.

Case iv) (r 2) (1 1 1 1 1 1 1 1)

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Now all radius-1-centers must have weight 4. Because A(8,4,4) we are able to locate 14 centers and no more.

Th ' lS lna y proves f' 11 t h ' e optlma lty l ' 0 f 372 • • 914 156 •

14 (cf. [5])

Vlo. 37.919.148• Let the radius-2-center be chosen in (1 1 1 1 1 1 1 1). Then the radius-1-centers all must have weight ~ 4. Choose one of these centers in (0 0 0 0 0 0 0 0): The others must have weight 3 or 4. We add a parity bit and observe that A(9,4,4)

=

18 (cf. [5]). So apparently 19

radius-1-cen-ters can be located altogether. Since A(8,3) = 20 (cf. [5]) we cannot locate 20 centers besides the radius-2-center.

20 76

VII. 9 .1 . A consequence of the fact that A(8,3)

=

20 (cf. [5]).

Those packings of Hamming spheres in {O,l}S that are not optimal, are called suboptimal.

7 Altogether there are 15 optimal and 87 suboptimal combinations (mi)O'

4. Counting types by combinations in a period box in~8' An inductive principle Referring to the classification of combinations introduced in section 3 we form the matrix M as follows

Left-multiplication of this matrix equality by the row vector

[-3,5,49,105,-105,-49,-5,3J transforms its left hand side into an obviously nonnegative expression (the optimal combinations give a nonnegative, the sub-optimal combinations even a positive contribution each), so that apparently on the right hand side for the inventory of types we must have

Let us now choose for V the set of vertices within a period box of a periodic tiling. Because of the equicontribution of all orthants of the cubistic cross-polytopes to the inventory of types in a period box the above inequality re-duces to a necessary condition for the existence of a periodic tiling inm

S' Le.

where q. := the number of vertices of type i in one orthant of the cubistic 1.

cross-polytopes.

Remark. The choice of the special row vector for left-multiplication was made according to a linear programming argument in order to get a strong

nonexis-tence result.

It should be kept in mind that the arguments in this section can be used only inE

S' but refer to ~e~io~ tilings with cubistic cross-polytopes, irrespec-tive of the fact, whether these polytopes are congruent or not.

This is very important, since a periodic tiling with congruent cubistic cross-polytopes inE (n ~ S) induces periodic tilings with the same period, but

n

with not necessarily congruent cubistic cross-polytopes in an ensemble of 8-spaces _ Those 8-spaces namely, that are obtained by keeping (n - S) integer-valued coordinates inm fixed. We only have to check how many 8-dimensional

n

cubistic cross-polytopes of different sizes are induced by a given radius-e cubistic cross-polytope in En (n ~ 8) in order to get a correct value for gi in the last inequality_ (see e.g. figure 3)_

To be more specific: Every radius-e cubistic cross-polytope inE induces n

Bn-s,s cubistic cross-polytopes of radius .e-s in 8-space (0:'; s :,; e). This mo-difies the generating function of the number of vertex points of type

\~

0

(~)

l.J= J pro orthant in the induced 8-dimensional sections of a radius-e polytope in n-space to the form

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00

I

e=O

g _n,e,tz e zt(1 + 2) n-8 _n

(1 - z)

Now i t is interesting to analyze the generating function of the expression in the quantities g1, ... ,g255' the nonnegativity of which is a necessary con-dition for the existence of a tiling.

This function has the form

7 6 2 5 3 4 (1 + z) n-8 [3 (1 - z ) - 5 (z - z ) - 49 (z - z ) - 105 (z - z ) ]"':""'--'--n (1 - z) 6 4 2 n-8 (1 +z) + 10 (1 +z) -4 (1 +z) + 1J(1 +z) [-4 6 4 ~ 2 n-7

=

(1-z) (1-z) (1-z) (1-z) -4S n_2(z) + lOS n-4(z) - 4S n-6(z) + S n-8(z) •

Hence, a necessary condition for the existence of a periodic tiling ofm

n

(n ~ 8) with radius-e cubistic cross-polytopes is

-4S _{n- ,e}2 + lOS _{n- ,e}4 - 4S

+

S ~ 0 •

n-6,e n-8,e

5. Computer results. Asymptotic estimates

In this section the last inequality of section 4 is applied.

For 8 ~ n ~ 50 i t was found by computer that for a periodic tiling with

ra-dius-e polytopes e < 0,37n - 0,71 (so p ,~ O,37n - 0,71. For large nand e asymptc

. n

tic considerations justify that the successive volumes in this inequality are successive terms in a geometric progression. The equality

6 4 2 2

-4a + lOa - 4a + 1 ~ 0 is violated by a ~ 2.07638. Now the recursive

relation for S (cf. section 2) implies nonexistence when ~ ~ 0.3735

n,e n

(n + 00). Hence p

In

5 0.3735 (n + (0). n

References

[1] ASTOLA, J . : On the nonexistence of certain perfect

Lee-error-correct-ing codes, Ann. Univ. Turku. Ser. A I 167 (1975), 1-13.

[2J ASTOLA, J.: On perfect codes in the Lee-metric. Ann. Univ. Turku. Ser.

A I 176 (1978), 1-56 (thesis).

[3J BASSALYGO, L.A.: A necessary condition for the existence of perfect

codes in the Lee metric. Mat. Zametki (Russian) 15 (1974), 313-320.

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[5J BEST, M.R., and A.E. BROUWER, J. MACWILLIAMS, A.M. ODLYSKO, N.J.A. SLOANE. Bounds for binary codes of length less than 25. IEEE Trans. IT 24

( 1978), 81-93.

[6J GOLOMB, S.W. and L.R. WELCH, Algebraic coding and the Lee-metric, in

"Error Correcting Codes" (H.B. Mann), 175-194, Wiley New York (1968). [7J LENSTRA, Jr., H.W. Necessary conditions for the existence of perfect

Lee-codes. Math. Centre Report ZN 59/75, Amsterdam (1975).

-[8J MULLER, P.: Das Existenzproblem perfecter Codes fur die Lee-metrik. Diplomarbeit, Kiel (1976).

[9J POST, K.A. Nonexistence theorems on perfect Lee-codes over large alpha-bets. Information and Control ~ (1975), 369-380.

[10J POST, K.A. Perfect 2-Lee error correcting codes over alphabets of size 5 or more do not exist for word length 5 or 6. Memorandum, Techno-logical University of Eindhoven, 1975-10 (1975).

[11J SPENCER, J. Maximal consistent families of triples. J. Comb. Theory 5 (1968), 1-8.

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n.· 2 n - 3 e -

a

D

e -

a

e .. I e • I e - 2 e co 3

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Figure 3. Dissecting a 3 - dimensional cubistic cross-polytope into "solid" 2 - dimensional cubistic cross-polytopes.