Testing sets for 1-perfect codes

S.V. Avgustinovich, A.Yu.Vasil’eva

Sobolev Institute of Mathematics, Koptyug pr. 4, Novosibirsk, 630090, Russia,avgust@math.nsc.ru, vasilan@math.nsc.ru

This paper continues the researsh of [1,2]. In [1] it was shown that a 1-perfectcode is uniquely determined by its vertices at the middle levels of hypercubeand in [2] the concerned formula was obtained. Now we prove that the verticesat the r-th level, r ≤ (n− 1)/2, of such a code of length n uniquely determineall code vertices at the lower levels.

1. Main result.

We denote n-dimensional vector space over GF (2) by En and call it a hyper-cube. We consider Hamming metric in En, i. e. the distance ρ(x,y) betweenvertices x and y of hypercube is equal to the number of positions in which thevertices differ. Hamming weight wt(x) of vertex x is equal to the number ofnonzero positions of x. Denote by Sr(x) (Br(x)) a sphere (a ball) of radius rwith the center x. The sphere Sr(0) = Wr centered in the all-zero vertex 0 iscalled a r-th level of hypercube.

A ϑ-centered function f : En → R is a function such that the sum of its valuesin a ball of the radius 1 is equal to ϑ. A perfect binary single-error-correctingcode C (briefly a 1-perfect code) of length n is a subset of En such that theset {B1(x) | x ∈ C} is a partition of En. The characteristic funstion of a1-perfect code is 1-centered.

Let Φ be a family of real functions over En and A, B ⊆ En. We call A aB-testing set for the family Φ if for any f, g ∈ Φ the following condition holds:

if f(x) = g(x) for any x ∈ A then f(x) = g(x) for any x ∈ B;.

in the other words if one knows the values of an arbitrary function f ∈ Φ overA then one can find all values of f over the set B.

Electronic Notes in Discrete Mathematics 21 (2005) 233–236

1571-0653/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2005.07.032

Let Φϑ be the family of all ϑ-centered functions over En. The main result ofthis paper is

Theorem 0.1. Let ϑ be real, v ∈ En and r ≤ (n − 1)/2. Then the Sr(v) isthe Br(v)-testing set for the family Φϑ.

The next Theorem can be easily get from the previous Theorem.

Theorem 0.2. Let v ∈ En and r ≤ (n − 1)/2. The set of vertices of a 1-perfect code of weight less than r is uniquely determined by the set of all codevertices of weight r.

2. Local spectra.

A k-dimensional face γ of the hypercube is the set of all vertices with fixed n−kcoordinates. Fix a vertex x ∈ γ. The local spectrum of a function f : En → Rin the face γ with respect to the vertex x is the (k + 1)-dimensional vector

l(x) = (l0(x), l1(x), . . . , lk(x)), where li(x) =∑

y∈γ, ρ(x,y)=i

f(y).

We can consider the partial ordering in En: x � y if xi ≤ yi, i = 1, . . . , n,where x = (x1, . . . , xn) and y = (y1, . . . , yn). For a vertex x of hypercube wedenote

γx = {y ∈ En | y � x}, γ⊥x = {y ∈ En | y � x}.

Pares of faces like this are called orthogonal. Note that dim γx = wt(x)and dim γ⊥

x = n − wt(x). Denote by l(x) = (l0(x), . . . , lwt(x)(x)) the localspectrum of a function f in the face γx with respect to x and by l⊥(x) =(l⊥0 (x), . . . , l⊥n−wt(x)(x)) the local spectrum of a function f in the face γ⊥

x withrespect to the vertex x.

The local spectra of a perfect code in two orthogonal faces with respect totheir common vertex were proved [5] to be in the tight interdependence. Moregenerally, it holds

Theorem 0.3. Let f be a 0-centered function, x ∈ Wi, i ≤ (n− 1)/2. Thenfor any j, 0 ≤ j ≤ n − i, it holds

l⊥j (x) =∑

q+r=j

(−1)qpirlq(x),

where pi2t = −pi

2t+1 = (−1)t

⎛⎝ (n − 1)/2 − i

t

⎞⎠ .

S.V. Avgustinovich, A.Yu. Vasil’eva / Electronic Notes in Discrete Mathematics 21 (2005) 233–236234

The proof of this theorem repeats almost literally the proof of its special caseconcerned perfect codes [5]. Now we are ready to prove Theorem 0.1 in thethe case ϑ = 0.

3. Proof of Theorem 0.1.

Let ϑ = 0 and f be an arbitrary 0-centered function over En. Without loss ofgenerality we consider v = 0. We will prove that the r-th level of hypercube isthe Dr-testing set for 0-centered functions, where Dr = W0

⋃W1

⋃. . .

⋃Wr.

We will show by induction on i that r-th level Wr of hypercube is Wi-testingset for 0-centered functions.

The base of induction: i = 0. Analogously with the case of 1-perfect codes[3,4] it can be obtained that

∑a∈En, wt(a)=r

f(a) = p0rf(0)

Thus f(0) is uniquely determined by the values of the function f over Wr.

The step of induction. Suppose that Wr is proved to be the Di−1-testing setfor 0-centered functions. Let x ∈ Wi, i = 1, . . . , r − 1. ¿From Theorem 0.3we have:

l⊥r−i(x) =r−i∑q=0

(−1)qpir−i−qlq(x),

f(x) = l0(x) =1

pr−i

(l⊥r−i(x) −

r−i∑q=1

(−1)qpir−i−qlq(x)

).

Here pr−i is nonzero and

l⊥r−i(x) =∑

a�x, wt(a)=r

f(a), lq(x) =∑

y�x, wt(y)=i−q

f(y).

By induction supposition the last sum is uniquely determined by the valuesof f over the r-th level. Hence f(x) is also uniquely determined by the valuesof f over the r-th level and Wr is the Di-testing set. Theorem 0.1 is proved.

References

[1] S.V. Avgustinovich, On a property of perfect binary codes, Discrete Analysisand Operation Research (in Russian), Vol. 2, No. 1, 4-6, 1995.

S.V. Avgustinovich, A.Yu. Vasil’eva / Electronic Notes in Discrete Mathematics 21 (2005) 233–236 235

[2] S.V. Avgustinovich and A. Y. Vasil’eva, Reconstruction of centered functions byits values on two middle levels of hypercube, Discrete Analysis and OperationResearch (in Russian), Vol. 10, No. 2, 3-16, 2003.

[3] S.P. Lloyd, Binary block coding, Bell Syst. Techn. J., Vol. 36. No. 2, 517-535,1957.

[4] H.S. Shapiro and D.L. Slotnick, On the mathematical theory of error correctingcodes, IBM J. Res. Develop. 3., No. 1, 25-34, 1959.

[5] A.Y. Vasil’eva, Local spectra of perfect binary codes, Discrete Analysis andOperation Research (in Russian), Vol. 6, No. 1, 16-25, 1999.

S.V. Avgustinovich, A.Yu. Vasil’eva / Electronic Notes in Discrete Mathematics 21 (2005) 233–236236