On the metrical rigidity of binary codes

S.V. Avgustinovich, a;1 F.I. Solov'eva a;2

aSobolev Institute of Mathematics, pr. Koptyuga 4, Novosibirsk 630090, Russia

Abstract

A code C in the n-dimensional vector space En over GF (2) is called metrically rigid

if every isometry I : C ! En with respect to the Hamming metric is extendable

to an isometry of the whole space En. A code C is reduced if it contains the all-

zero vector. For n large enough the metrical rigidity of length n reduced binary

codes containing a 2-(n; k; �)-design is proved. The class of such codes includes

all the families of uniformly packed codes of su�ciently large length satisfying the

condition d� � � 2; where d is the code distance and � is the covering radius.

Key words: Metrically Rigid Codes; Strongly Rigid Codes; 2-(n; k; �)-design;

Uniformly Packed Codes

1 Introduction

The aim of this paper is to investigate the problem of metrical rigidity of binary

codes. A notion of the metrical regidity is closely and naturally connected with

a well known in geometry notion of rigidity (see, for example, [1{3]). It should

be noted that the term rigid code is used in [4,5] in a di�erent context for

codes with the trivial automorphism group.

Let En be the n-dimensional vector space over GF (2) with respect to the

Hamming metric. A binary code C of length n is a subset of En. In what

follows, we omit the term binary because we only deal with such codes. A

mapping I : C ! En is called an isometry from the code C to the code

1 This research was partially supported by the Russian Foundation for Basic Re-

search under the grant 00-01-00916 and by INTAS 97-1001.

E-mail address: avgustl@math.nsc.ru (S.V. Avgustinovich)2 This research was supported by the Russian Foundation for Basic Research under

the grant 00-01-00822 and partially supported by the grant FZP \Integration" 274.

E-mail address: sol@math.nsc.ru (F.I. Solov'eva)

Preprint submitted to Elsevier Preprint 4 April 2001

I(C) if d(x; y) = d(I(x); I(y)) for all codewords x; y 2 C: A code C in En

is called metrically rigid if every isometry I : C ! En with respect to theHamming metric is extendable to an isometry of the whole space En (see [6]).The automorphism group of En is transitive on En, therefore without loss ofgenerality for an isometry I : C ! En we can put 0n 2 C and I(0n) = 0

n,where 0n is the all-zero vector of length n. We shall call an isometry I fromthe code C into the code I(C) reduced if it preserves weights of codewords.It is clear that an isometry I such that I(0n) = 0

n is reduced. Two codesC;D � En are equivalent if there exists an isometry of En which transformsthe code C into the code D. It is known that every isometry is de�ned by amapping Av

�: x ! �(x) + v; where � is a permutation of coordinates and

v 2 En:

Let R be any metric space with the automorphism group rich enough, for ex-ample double transitive. Two metric subspaces R1 and R2 in R are equivalentif there exists an automorphism I from the automorphism group of R suchthat I(R1) = R2: Quite often all problems concerning metric spaces are inves-tigated up to equivalence. Let us consider the situation when only metricalstructure of the metric subspace is important, but not the way how a metricsubspace puts in R. The investigation of the situation from equivalence pointof view is not possible. The isometric approach permits the di�culties to beovercome. The situation described above takes place both in geometry (see[1-3, 7]) and discrete mathematics (see [6, 8-11]).

If two codes are equivalent they are isometric, but the inverse is not true inevery case. For example the codes

C = f(0000); (1100); (1010); (0110)g

and

D = f(0000); (1100); (1010); (1001)g

are isometric but are not equivalent. All Hadamard codes obtained fromHadamard matrices by replacing -1 to 1 and 1 to 0 are mutually isometricbut there are a lot of nonequivalent Hadamard matrices (see [15]) and there-fore Hadamard codes.

A code C is called strongly rigid if it is metrically rigid and every code ofthe same length containing C is also metrically rigid. In the paper we onlyconsider reduced codes, i.e. codes with 0 2 C. We call a 2-(n; k; �)-designwith the joined all-zero word 0

nreduced design. It is proved that any reduced

2-(n; k; �)-design is strongly rigid for n large enough. The class of stronglyrigid codes containing reduced 2-(n; k; �)-designs includes all the families ofuniformly packed codes of su�ciently large length satisfying the conditiond � � � 2; where d is the code distance and � is the covering radius. It hasbeen shown in [12-14] that uniformly packed codes contain (d��)-designs and

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include BCH-codes of distance 5 and 7, Preparata codes, Goethals codes of

distance 7, extended perfect codes.

It should be noted that it is more di�cult to prove rigidity for codes with

large code distance than for codes with small distances.

The metrical rigidity of the following classes of codes has been established in

[9,6,10]:

1) perfect q-ary codes for q � 2; with the exception of the binary Hamming

code of length 7 and the ternary Hamming code of length 4;

2) the binary even-weight code of length n with the exception of the case of

n = 4;

3) q-ary (n; n � 1) MDS codes with the exception of several codes of small

length;

4) full constant-weight codes.

2 Metrical rigidity of codes

Let N = f1; 2; : : : ; ng. A subset D � En of weight k vectors is called a 2-

(n; k; �)-design if the number of vectors in D with ones in the ith and jth

coordinates is equal to exactly � for all di�erent i; j 2 N . The set supp(x) =

fk : xk = 1g is called the support of a vector x = (x1; : : : ; xn) 2 En.

Let i 2 N . We call a set Si � En a �-homogeneous i-star if the following

conditions hold:

1) wt(v) = k for every v 2 Si;

2) i 2 supp(v) for every v 2 Si;

3) the number of vectors in Si with ones in the jth position is equal to � for

every j 2 N; j 6= i.

Is it easy to see from the de�nition of a 2-(n; k; �)-design and the above de�-

nition that the design contains a �-homogeneous i-star for every i 2 N . The

i-star is uniquely de�ned by the design.

We call a set S�

i� E

n a u�y i-star if the following conditions hold:

1) wt(v) = k for every v 2 Si;

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2) supp(v) \ supp(u) = fig for all v; u 2 S�

i; v 6= u;

3) jS�

ij > k

2 � k + 1:

Note that the condition 2 implies d(v; u) = 2k � 2.

An incidence matrix of a vector set S is a rectangular array whose lines are

vectors from S.

Using the de�nition of a 2-(n; k; �)-design it is not di�cult to prove the fol-

lowing

Proposition 1 For every �-homogeneous i-star Si; i 2 N; we have

jSij = �(n� 1)=(k � 1):

Proposition 2 For all n � k4 and i 2 N , every �-homogeneous i-star Si

contains a u�y i-star S�

i:

Proof. Let us consider a 2-(n; k; �)-design and for every i 2 N take the

maximal cardinality set S�

iof its of vectors with ones in ith coordinate and

pairwise distances 2k � 2. We should prove that S�

iis the u�y i-star, i.e.,

that jS�

ij > k

2 � k + 1.

Using the de�nition of �-homogeneous i-star Si we conclude that S�

i� Si:

Let us take the incidence matrix of Si and delete the ith colomn, all colomns

containing zero-colomns of the submatrix S�

iand all rows containing the sub-

matrix S�

iin it. On the one hand, the number of ones in all colomns of the

matrix obtained equals to jS�

ij(k� 1)(�� 1); and on the other hand, there are

not less than jSij � jS�

ij ones in the rows of the matrix . Therefore

jSij � jS�

ij � jS

�

ij(k � 1)(�� 1):

>From this inequality, we get using Proposition 1 that

�(n� 1)=(k � 1) � jS�

ij((k � 1)(�� 1) + 1);

i.e.

n �jS�

ij(k � 1)((k � 1)(�� 1) + 1)

�+ 1:

If jS�

ij � k

2 � k + 1; then

n �(k2 � k + 1)(k � 1)((k � 1)(�� 1) + 1)

�+ 1 < k

4;

4

a contradiction.

}

Proposition 3 If n � k4, then every reduced isometry I takes every u�y

i-star into the u�y i0-star for some i

0 2 N:

Proof. Suppose that the proposition is not true and the image I(S�

i) of the

u�y i-star S�

iis not a u�y i

0-star for any i

0 2 N:

Without loss of generality, � = (1; : : : ; 1; 0; : : : ; 0) 2 I(S�

i). Since a reduced

isometry preserves weights, every vector in I(S�

i) has weight k. Suppose that

there are is � 0 vectors having ones in the sth coordinate, s = 1; : : : ; k: The

assumption that I(S�

i ) is not a u�y i-star implies existence of at least two

is > 0.

Eliminating the vector � in I(S�

i ), we have

kX

s=1

is = jI(S�

i )j � 1:

Compare a vector having one in the sth coordinate, s � k, with a vector �

having one in the jth coordinate, s 6= j; where is; ij > 0: A reduced isometry

preserves an intersection of vectors, i.e. preserves a number of common 1th

coordinates in these vectors. Then

jsupp(�) \ supp( )j = jsupp(�) \ supp(�)j = jsupp( ) \ supp(�)j = 1:

Hence only coordinates s and �j are equal to 1 among the �rst k coordinates

of vectors and �, and there is exactly one coordinate �t such that t > k and

t = �t = 1 among coordinates of the vector �. Therefore is � k�1, ij � k�1

and

jI(S�

i )j � 1 =

kX

s=1

is � k(k � 1):

Hence

jI(S�

i )j � k2 � k + 1:

By the de�nition of an isometry we have jI(S�

i )j = jS�

i j. But jS�

i j > k2�k+1;

a contradiction.

}

Proposition 4 If n � k4, then every reduced isometry I takes every �-

homogeneous i-star into a �-homogeneous i0-star for some i0 2 N:

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Proof. Let us consider the p � n incidence matrices M and M0 of the stars

Si and I(Si) respectively, where Si is a �-homogeneous i-star and p = jI(Si)j:

By the de�niton of an i-star, every row of the matrix M contains k ones and

every column except the ith one contains � ones. Every reduced isometry I

preserves weigths, therefore there are k ones in every row of the matrix M0.

Using Proposition 3 and contradiction arguments, it is easy to prove that there

exists some i0 2 N such that the i0th column in the matrix M0 contains only

ones.

Let �1; �2; : : : ; �i0�1; �i0+1; : : : ; �n be numbers of ones in all columns except the

i0th column of the matrixM 0. We shall prove that �j = � if j 6= i

0: Let us take

the sum � of all pairwise distances between vectors from Si: It is clear that

a reduced isometry preserves the sum of the distances. Taking into account

that every \zero-one" pair in a column of the matrix M (and similarly of the

matrix M0) adds one into the sum �, we obtain that

� = (n� 1)�(p� �) =X

j 6=i0

�j(p� �j) = pX

j 6=i0

�j �X

j 6=i0

�2

j :

ThusP

j 6=i0 �j = (n� 1)�. According to the Cauchy-Bunyakowsky inequality,

we have X

j 6=i0

�2

j �1

(n� 1)(X

j 6=i0

�j)2

and the equality holds only if �j = � for all j 2 N; j 6= i0.

}

Let us consider a 2-(n; k; �)-design and its reduced isometry I. According

to Proposition 4, the pair (Si; I) de�nes a mapping � : N ! N such that

�(i) = i0.

Using contradiction arguments and Proposition 4, it is not di�cult to prove

the following

Proposition 5 The mapping � is a permutation.

Proposition 6 Let a code C contain a 2-(n; k; �)-design. Let u be a vector in

C. Then for every reduced isometry I we have

1) if i 2 supp(u) then �(i) 2 supp(I(u));

2) if i =2 supp(u) then �(i) =2 supp(I(u)):

Proof. Let J� : En ! En be the isometry generated by the permutation

�. For both cases let us count the sum of distances between any vector u =

(u1; : : : ; un) 2 C and all vectors of the �-homogeneous i-star Si(C). If the

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sums are di�erent it means according to Propositions 4 and 5 that a reduced

isometry I is extendable to the isometry J�.

There are two cases.

1. Let i 2 supp(u) and w(u) = t. Then ui= 1 and the vector u gives zero to the

sum of the distances. There are t� 1 coordinates of the vector u equal to one.

Every such coordinate of u together with every vector v 2 Si(C) having zero in

the ith coordinate gives one in the sum of the distances. There are jSi(C)j��

such vectors in Si(C). There are n � t zero coordinates of the vector u and

every such coordinate together with every vector v 2 Si(C) having one in the

ith coordinate gives one in the sum of the distances. There are � such vectors

in Si(C). Summing up all distances we get

X

v2Si(C)

d(u; v) = (t� 1)(jSi(C)j � �) + (n� t)�: (1)

2. In the case i =2 supp(u) considerations analogous to those in the case 1 allow

to obtain

X

v2Si(C)

d(u; v) = jSi(C)j+ t(jS

i(C)j � �) + (n� t� 1)�: (2)

Comparing (1) and (2), we see that the numbers are di�erent.

}

It follows from Propositions 2-6 that the automorphism � coincides with the

reduced isometry I. Hence every reduced 2-(n; k; �)-design is metrically rigid.

It is not di�cult to prove that every reduced code of length n; n � k4; contain-

ing 2-(n; k; �)-design is metrically rigid. Therefore every reduced 2-(n; k; �)-

design is strongly rigid and we obtain the main result of the paper.

Theorem 7 For n � k4 every reduced 2-(n; k; �)-design is a strongly rigid

code.

References

[1] V.A. Aleksandrov, An example of a one-dimensional rigid set in the plane,Siberian Math. J. 34 (6) (1993) 999-1004.

[2] I. Herburt, Rigidity of products, Geom. Dedicata 46 (1993) 243-248.

7

[3] I. Herburt, S. Ungar, Rigid sets of dimension n � 1 in Rn, Geom. Dedicata 76

(1999) 331-339.

[4] H. Oral, K.T. Phelps, Almost all self-dual codes are rigid, J. Comb. Theory A

60 (2) (1992) 264-276.

[5] H. Lefmann, K.T. Phelps, V. Roedl, Rigid linear binary codes, J. Comb. TheoryA 63 (1) (1993) 110-128.

[6] F. I. Solov'eva, S.V. Avgustinovich, T. Honold, W. Heise, On the extendabilityof code isometries, J. of Geometry 61 (1998) 3-16.

[7] G. Laman, On graphs and rigidity of plane skeletal structures, J. Engrg. Math.

4 (1970), 331-340.

[8] G.A. Kabatyanskij, V.I. Levenstein, Bounds on packing on a sphere and in space,Problem of Inform. Transm. 14 (1) (1978) 1-17.

[9] S.V. Avgustinovich, On isometry of perfect binary codes, Proc. of Institute of

Math. SB RAN 27 (1994) 3-5.

[10] F.I. Solov'eva, S.V. Avgustinovich, T. Honold, W. Heise, Metrically rigid codes,Proc. Sixth Int. Workshop on Algebraic and Comb. Coding Theory. Pskov,

Russia. September (1998) 215-219.

[11] S.V. Avgustinovich, On strong isometry of binary codes, Discrete Anal. and

Oper. Research 1 (7) 3 (2000) 3-5 (in Russian).

[12] N.B. Semakov, V.A. Zinov'ev, G.V. Zajzev, Uniformly packed codes, Problemsof Inform. Transm. 7 (1) (1971) 38-50.

[13] L.A. Bassalygo, G.V. Zajzev, V.A. Zinov'ev, On uniformly packed codes,Problems of Inform. Transm. 10 (1) (1974) 9-14.

[14] L.A. Bassalygo, V.A. Zinov'ev, Remark on uniformly packed codes, Problemsof Inform. Transm. 13 (3) (1977) 22-25.

[15] M. Hall, Jr., Combinatorial Theory (Waltham (Massachusetts), Toronto-London, 1967).

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