On metric rigidity for some classes of codes

ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 1, pp. 15–27. c© Pleiades Publishing, Inc., 2011.Original Russian Text c© D.I. Kovalevskaya, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 1, pp. 19–32.

CODING THEORY

On Metric Rigidity for Some Classes of Codes

D. I. Kovalevskaya

Sobolev Institute of Mathematics,Siberian Branch, Russian Academy of Sciences, Novosibirsk

[email protected]

Received April 23, 2010; in final form, December 10, 2010

Abstract—A code C in the n-dimensional metric space Fnq over the Galois field GF (q) is said to

be metrically rigid if any isometry I : C → Fnq can be extended to an isometry (automorphism)

of Fnq . We prove metric rigidity for some classes of codes, including certain classes of equidistant

codes and codes corresponding to one class of affine resolvable designs.

DOI: 10.1134/S0032946011010029

1. INTRODUCTION

By Fnq we denote the n-dimensional metric space over the Galois field GF (q) equipped with the

Hamming metric. A code of length n is an arbitrary subset of the metric space Fnq . Parameters of a

q-ary code C are denoted by (n,M, d)q , where n is the codeword length, M is the code cardinality,and d is the code distance; for a binary code we skip the notation q = 2 and write Fn for Fn

2 . By 0n

and 1n we denote the all-zero and all-one vectors of length n, respectively.

A map I : C → C ′, where C and C ′ are codes of the same cardinality in Fnq , is called an isometry

from C to I(C) = C ′ if d(x, y) = d(I(x), I(y)) for all codewords x, y in C.

It is known that the automorphism group of Fnq is confined to all isometries of Fn

q and is of theform

Aut(Fnq ) = {(π;σ1, . . . , σn) : π ∈ Sn, σi ∈ Sq, 1 ≤ i ≤ n},

where Sn and Sq are symmetric groups of orders n and q, respectively. The map (π;σ1, . . . , σn)acts on vectors of Fn

q as follows:

(π;σ1, . . . , σn)(x1, x2, . . . , xn) = π(σ1(x1), σ2(x2), . . . , σn(xn))

= π(z1, z2, . . . , zn)

= (zπ(1), zπ(2), . . . , zπ(n)),

for any vector (x1, x2, . . . , xn) in Fnq , where zi = σi(xi), i = 1, . . . , n. Two codes, C and D, in F

nq are

said to be equivalent if there exists an isometry of Fnq that takes C to D. The set of automorphisms

of Fnq that fix a code C is called the automorphism group of C and is denoted by Aut(C); i.e.,

Aut(C) = {(π;σ1, . . . , σn) ∈ Aut(Fnq ) : (π;σ1, . . . , σn)(C) = C}.

It is known that all isometries of Fn are given by maps of the form (π, v) : x �→ π(x)+v, where πis a permutation of coordinates and v is a vector in F

n; i.e., in the binary case the automorphismgroup of C is of the form

Aut(C) = {(π, v) : π(C) + v = C, π ∈ Sn, v ∈ Fn}.

15

16 KOVALEVSKAYA

The set of permutations of n coordinates that fix a code C is called the symmetry group of C andis denoted by Sym(C).

By Witt’s theorem [1], any linear isometry between two linear subspaces of any metric vectorspace can be extended to an isometry of the whole space.

There are several definitions of metric rigidity. A code C ⊂ Fnq is said to be weakly metrically

rigid if any code C ′ = I(C) is equivalent to C. In [2], weak metric rigidity of perfect binary codesof lengths greater than 15 is proved (in that paper, weak metric rigidity was called metric rigidity).

Following [3–5], we call a code C ⊂ Fnq metrically rigid if any isometry I : C → F

nq can be

extended to an isometry (automorphism) of Fnq .

We call a code C ⊂ Fnq metrically rigid in a narrow sense if for any isometry I : C → C there

exists an isometry I ′ of the whole space Fnq such that I|C = I ′|C .

If a code is not metrically rigid (weakly metrically rigid, metrically rigid in a narrow sense),we say that it is metrically nonrigid (weakly metrically nonrigid, metrically nonrigid in a narrowsense, respectively).

It follows from the above definitions that any metrically rigid code is weakly metrically rigidand metrically rigid in a narrow sense. The difference between the definitions of metric rigidity andweak metric rigidity can be observed, for instance, for the Hamming code of length 7. Indeed, theHamming code of length 7 is weakly metrically rigid, since it is unique up to equivalence. On theother hand, it is metrically nonrigid, since the cardinality of the set of isometries of the Hammingcode that fix the zero vector is greater than the cardinality of the symmetry group of this code.The difference between the definitions of metric rigidity and narrow-sense metric rigidity can beshown by the following example.

Example. Consider the binary code C = {0000, 1100, 1010, 1001}. One easily checks that everyisometry of C fixing the code can be extended to an isometry of the whole space F

4. Hence, thiscode is metrically rigid in a narrow sense. On the other hand, it is weakly metrically nonrigid, sincethere exists a code, for instance, D = {0000, 1100, 1010, 0110}, which is isometric but nonequivalentto C (see also Lemma 4).

In some situations, it is easier to work with the notion of narrow-sense metric rigidity ratherthan metric rigidity (see, e.g., Section 3).

Recall that a face of dimension k of the binary n-dimensional metric space Fn is a set

Bn;i1,...,in−kσ1,...,σn−k = {(v1, . . . , vn) ∈ F

n : vi1 = σ1, . . . , vin−k= σn−k},

where σ1, . . . , σn−k are fixed constants in the set {0, 1}. By dimΓ(C) we denote the face dimen-sion of C, i.e., the dimension of the smallest face of the metric space that contains C. A mapI : C → I(C), where C and I(C) are codes of the same cardinality in F

n, is said to be a strongisometry if for any C∗ ⊆ C we have

dimΓ(C∗) = dimΓ(I(C

∗)).

A code C in Fn is said to be uniformly packed in a wide sense (see [6]) if there exist numbers

α0, . . . , αρ such that for any vector x ∈ Fn we have the equality

ρ∑

i=0αifi(x) = 1, where fi(x) is the

number of codewords at distance i from x and ρ = maxy∈Fn

d(y,C) is the covering radius of C.

A code C in Fnq is called an (n, k) MDS code if d = n − k + 1, where k = logq M , i.e., if the

parameters of the code meet the Singleton bound. A code with distance d is said to be equidistantif the distance between any two codewords is the same and equals d (see [7, ch. 15]). An equidistantcode with parameters (n,M, d)q is optimal if d = nM(q − 1)/(M − 1)q.

PROBLEMS OF INFORMATION TRANSMISSION Vol. 47 No. 1 2011

ON METRIC RIGIDITY 17

A balanced incomplete block design with parameters (v, b, r, k, λ), also referred to as a 2-(v, k, λ)-design, is an arrangement of v distinct elements in b blocks such that each block contains exactly kdifferent elements, each element occurs in exactly r different blocks, and each unordered pair ofdistinct elements occurs in exactly λ blocks. Then the relations

bk = vr, r(k − 1) = λ(v − 1)

hold. To any design with parameters (v, b, r, k, λ) we can assign a binary code with parameters(v, b, d) for some d ≤ 2k (here to any block there corresponds a codeword in F

v with ones in thepositions that correspond to the elements contained in this block). A parallel class in a 2-(v, k, λ)-design, where v ≡ 0 (mod k), is a set of v/k pairwise disjoint blocks. A resolvable design is a2-design where the set of blocks can be partitioned into r disjoint parallel classes. A resolvabledesign is said to be affine resolvable if the intersection of any two blocks from different parallelclasses have the same number of elements.

2. SOME KNOWN RESULTS ON METRIC RIGIDITY OF CODES

It is well known that some binary Hadamard codes of length n ≥ 16, which are obtained fromHadamard matrices of the same order by replacing 1 with 0 and −1 with 1, are metrically non-rigid, since there exist nonequivalent such matrices and, correspondingly, nonequivalent Hadamardmatrices.

The following facts were proved in [3]:

(a) All perfect q-ary codes are metrically rigid, except for the binary Hamming code of length 7and the ternary Hamming code of length 4;

(b) All q-ary (n, n − 1) MDS codes are metrically rigid, except for two codes of length 3 and onecode of length 4;

(c) All q-ary (q, 2) and (q + 1, 2) MDS codes are metrically nonrigid, except for (2, 2) and (3, 2)MDS codes;

(d) A binary linear code with parameters (n, 2n−1, 2) is metrically rigid if and only if n = 4.

In [4] it was proved that full constant-weight q-ary codes are metrically rigid. In [5] it was foundthat for n ≥ k4 each binary code containing a 2-(n, k, λ)-design and the zero vector is metricallyrigid. Since any t-(v, k, λ)-design, t ≥ 2, is a 2-(v, k, λ′)-design, any code with the zero vectorcontaining a t-(v, k, λ)-design is metrically rigid. Therefore, all extended primitive BCH codes andextended perfect codes are metrically rigid. This property is also possessed by uniformly packedcodes satisfying the condition d− ρ ≥ 2, where d is the code distance and ρ is the covering radius;the latter include such codes as BCH codes with distances 5 and 7, wide-sense Preparata codes,wide-sense Goethals codes with distance 7 (see [5]), and also the corresponding extended codes.

In [8], optimal equidistant codes of length n =q2μ− 1

q − 1and cardinality M = q2μ with code

distance d = qμ were studied, where μ is a positive integer; it was proved that an optimal equidis-tant code with these parameters exists if and only if there exists an affine resolvable design withparameters (v, b, r, k, λ), where

v = q2μ, b =q3μ− q

q − 1, r =

q2μ− 1

q − 1, k = qμ, λ =

qμ− 1

q − 1.

In [9] it was proved that q-ary equidistant codes of length n = q and cardinality M = (q− 1)2 withcode distance d = q − 1 exist if both q and q − 1 are prime powers.

In the present paper we demonstrate that optimal q-ary equidistant codes with parameters(q2μ− 1

q − 1, q2μ, qμ

)

qfor q ≥ 3 and μ ≥ 1, q-ary equidistant codes with parameters (q, (q−1)2, q−1)q


18 KOVALEVSKAYA

for q ≥ 5, binary equidistant codes with parameters (n, [2n/d], d) such that

√16 + 4n − 4 < d ≤ n/2, d ≡ 0 (mod 2),

and also codes corresponding to the class of affine resolvable designs with parameters (n, qs, s, qμ, λ),where

n = q2μ, q ≥ 3, s ≥ ncμq , cμq = log2 n/ log2(q!n),

are metrically nonrigid. Furthermore, we investigate weak metric rigidity of some other codes, suchas the first-order Reed–Muller codes, Nordstrom–Robinson code, and some classes of equidistantcodes from [10].

3. METRIC RIGIDITY OF THREE CLASSES OF EQUIDISTANT CODES ANDOF CODES CORRESPONDING TO AFFINE RESOLVABLE DESIGNS

To prove Theorems 1 and 2, we prove several auxiliary lemmas and use the notion of metricrigidity in a narrow sense.

Metric nonrigidity of the classes of equidistant codes and codes corresponding to affine resolvabledesigns is easier to prove in a narrow sense. Recall that it is easily seen from the definitions that ametrically rigid code is also metrically rigid in a narrow sense. Hence it follows that any narrow-sense nonrigid code is metrically nonrigid.

Let ISO(C) be the subset of isometries of a code C that fix this code:

ISO(C) = {I : C → C}.

Then we have the following fact.

Lemma 1. If the cardinality of the subset of all isometries of an arbitrary q-ary code C thatfix the code is greater than the order of the automorphism group of this code, then C is metricallynonrigid in a narrow sense.

Proof. Let Iso(C) be a subset of the set of isometries ISO(C) of a code C such that Iso(C) ⊆ISO(C) and |Iso(C)| > |Aut(C)|. Assume that every isometry of C from Iso(C) can be extended toan isometry of the whole space Fn

q . Since the number of isometries of C that fix the code is greaterthan the cardinality of the set of automorphisms of this code, there exists isometries I1, I2 ∈ Iso(C),I1 = I2, and an isometry I ′ of Fn

q such that I ′ is an extension of both I1 and I2, i.e., such that forall x in C we have I ′(x) = I1(x) and I ′(x) = I2(x). Hence, the action of I1 and I2 on all codewordsof C is the same; i.e., I1 ≡ I2. This contradicts the condition I1 = I2. Hence, not every isometryof C can be extended to an isometry of the whole space, and the code C is nonrigid in a narrowsense. �

Furthermore, there is the following statement, which ensures metric nonrigidity of an equidistantcode in the case where a code obtained from the original code by shortening is a metrically nonrigidequidistant code.

Proposition 1. Let an equidistant code C with parameters (n,m, d)q be metrically nonrigid ina narrow sense. Then every equidistant code with parameters

(N,M, d)q , N > n, M ≥ m

constructed from C by extension and/or adding new codewords is metrically nonrigid in a narrowsense.

Proof. Let C ′ be an equidistant code with parameters (N,m, d)q constructed from C by exten-sion only.



Since C is nonrigid in a narrow sense, there exists an isometry I : C → C that cannot beextended to an isometry of the whole space F

nq .

Let an isometry I ′ : C ′ → C ′ act on codewords of the extended code C ′ in the same way as I actson the corresponding codewords of the original code C. Then, obviously, I ′ cannot be extendedto an isometry of FN

q of the form σ = (π1, π2;σ1, . . . , σN ), where π1 ∈ Sn, π2 acts on the elements{n+1, n+ 2, . . . , N}, and σi ∈ Sq, 1 ≤ i ≤ N . We note that, since C and C ′ are equidistant codeswith the same distance d, each column with number from the set {n+1, . . . , N} in the code matrixof C ′ is of the form (a, . . . , a)T , a ∈ F

q. Therefore, if the code matrix of the original code C hasno columns of the form (a, . . . , a)T , a ∈ F

q, then the isometry I ′ cannot be extended to any of theremaining isometries of FN

q other than σ. If the code matrix of C contains columns of the form

(a, . . . , a)T , a ∈ Fq, then any isometry other than σ which could be an extension of I ′ leads to a

permutation of such columns in the code matrix of C ′. Let i1, . . . , it be numbers of the columns ofthe form (a, . . . , a)T in the code C, and let e ∈ Sq and e′ ∈ SN be identity permutations. If I ′ canbe extended to an isometry of FN

q , then the action of I ′ on C ′ is equivalent to sequential applicationof isometries Ψ1, Ψ2, and Ψ3 to the code C ′, where

• Ψ1 = (π;σ1, . . . , σn, e, . . . , e), π acts on the elements {1, . . . , n} \ {i1, . . . , it}, σj ∈ Sq, j ∈{1, . . . , n}, and σik = e, k ∈ {1, . . . , t};

• Ψ2 = (e′;σ′1, . . . , σ

′n, e, . . . , e), σ

′j ∈ Sq and all σ′

j = e, except for, maybe, σ′i1 , . . . , σ

′it ;

• Ψ3 = (e′; e, . . . , e, σn+1, . . . , σN ), σj ∈ Sq, j ∈ {n + 1, . . . , N}.However, there are no suitable isometries of the form Ψ1 and Ψ2, since the isometry I is notextendable to an automorphism of Fn

q . Therefore, the isometry I ′ of C ′ cannot be extended to an

automorphism of FNq .

Let D be an equidistant code with parameters (N,M, d)q constructed from C ′ by adding newcodewords, and let I ′′ : D → D be an isometry of D that acts on C ′ in the same way as I ′. If thereexists (π ∈ SN ;σ1 ∈ Sq, . . . , σN ∈ Sq), an extension of I ′′ to an automorphism of FN

q , then thisautomorphism is also an extension of the isometry I ′ for the subcode C ′ of D, a contradiction.Hence, the isometry I ′′ cannot be extended to an isometry of the whole space F

Nq , and the code D

is metrically nonrigid in a narrow sense. �Recall that for any prime power q there exists a q-ary equidistant code with parameters n = q+1,

M = q2, and d = q, and for any prime power q, provided that q − 1 is also a prime power, thereexists a q-ary equidistant code with parameters

n = q, M = (q − 1)2, d = q − 1.

Lemma 2. An equidistant code Cqq with the parameters (q, (q − 1)2, q − 1)q, q ≥ 5, where both

q and q − 1 are prime powers, is metrically nonrigid in a narrow sense.

Proof. Consider the set of isometries I : Cqq → Cq

q and the automorphism group of Fqq. Let us

show that the size of the set of isometries is greater than the order of the automorphism groupof Fq

q. By Lemma 1, this would imply that Cqq is nonrigid in a narrow sense.

Since n = q, we have|AutFq

q| = q! (q!)q = (q!)q+1.

Since Cqq is an equidistant code, we have

|ISO(Cqq )| = (q − 1)2!.

The following cases are possible.

Case 1. Let q = 5. Then

|ISO(Cqq )| = 16! > (5!)6 = (q!)q+1 = |AutFq

q|.


20 KOVALEVSKAYA

Case 2. Let q ≥ 7, where both q and q− 1 are prime powers. Compare the numbers (q!)q+1 and(q − 1)2!. We show that for any 2 ≤ k ≤ q − 1 we have the inequality

(kq − 2(k − 1)

)(kq − 2(k − 1)− 1

)> k2(k − 2)2. (1)

Indeed,

(kq − 2(k − 1)

)(kq − 2(k − 1)− 1

)− k2(k − 2)2

= k2q2 − q(4k(k − 1) + k

)+ 2(k − 1)(2k − 1)− k2(k − 2)2

= q2 − q(4− 3

k

)+ 2

(1− 1

k

)(2− 1

k

)− (k − 2)2.

Since 2 ≤ k ≤ q − 1, we have 4− 3

k≥ 4 +

3

1− qand 1− 1

k≥ 1

2. Therefore,

q2 − q(4− 3

k

)+ 2

(1− 1

k

)(2− 1

k

)− (k − 2)2 ≥ q2 − q

(4 +

3

1− q

)− (q − 3)2 + 2 · 1

2· 32

= 6q − 9− q(4 +

3

1− q

)+

3

2

= q(2 +

3

q − 1

)− 15

2.

Since q ≥ 7, we have 0 <3

q − 1≤ 1

2, 2q − 15

2> 0, and q

(2 +

3

q − 1

)− 15

2> 0. Hence we find

that inequality (1) is always valid for 2 ≤ k ≤ q − 1, where q ≥ 7.

It is convenient to use the following representation of the number (q − 1)2!:

(q − 1)2! =

(

22q−2∏

i=q+1

i

)(

33q−4∏

i=2q−1

i

)(

44q−6∏

i=3q−3

i

)(

55q−8∏

i=4q−5

i

)

× . . .

×(

(q − 1)

(q−1)q−2(q−2)∏

i=(q−2)q−(2(q−3)−1)

i

)(

q

q2−2(q−1)−1∏

i=(q−1)q−(2(q−2)−1)

i

)

. (2)

Let us separately consider the (k − 1)st factor in this representation for various values of k.

For k = 2, the product (q+1)(q+2) . . . (2q− 4)(2q − 3)(2q− 2) contains precisely q− 2 factors,each of them being greater than 2; therefore, taking into account that

q + 1 > q, 2q − 4 > q, q + 2 > 2, 2q − 3 > 4, 2q − 2 = 2(q − 1),

we obtain

(q + 1)(q + 2) . . . (2q − 4)(2q − 3)(2q − 2) > (4q)2 · 2q−7 · 2(q − 1) = (q − 1)q2 · 2q−2.

For k = 3, the product (2q − 1)2q . . . (3q − 6)(3q − 5)(3q − 4) contains precisely q − 2 factors,each of them being greater than 3; therefore, taking into account that

2q − 1 > q, (3q − 5)(3q − 4) > 9(q − 1), 2q > 9,

we obtain

(2q − 1)2q . . . (3q − 6)(3q − 5)(3q − 4) > 9q · 3q−6 · 9(q − 1) = q(q − 1) · 3q−2.



For 4 ≤ k ≤ q − 1, the product ((k − 1)q − 2k + 5) . . . (kq − 2(k − 1) − 1) × (kq − 2(k − 1))contains precisely q − 2 factors, each of them being greater than k; therefore, using inequality (1),we obtain

((k − 1)q − 2k + 5

). . .

(kq − 2(k − 1)− 1

)(kq − 2(k − 1)

)> nq−4n2(n− 2)2

= (n − 2)2nq−4.

Finally, for k = q, the product ((q−1)q− (2(q−2)−1)) . . . (q2−2(q−1)−2)× (q2−2(q−1)−1)contains precisely q − 3 factors, each of them being greater than q; therefore, using the obviousinequality

(q2 − 2(q − 1)− 1

)(q2 − 2(q − 1)− 2

)> q2(q − 2)2,

we obtain((q − 1)q − (2(q − 2)− 1)

). . .

(q2 − 2(q − 1)− 2

)(q2 − 2(q − 1)− 1

)> qq−5q2(q − 2)2

= (q − 2)2qq−3.

Hence, using factorization (2) for the number (q − 1)2!, we get

(q − 1)2! > 2(q − 1)q2 · 2q−2 · 3(q − 1)q · 3q−2 · 4 · 22 · 4q−2 · 5 · 32 · 5q−2 × . . .

× (q − 1)(q − 3)2(q − 1)q−2q(q − 2)2qq−3 = (q!)q+1

for q ≥ 7.

Thus, for all q ≥ 5, where both q and q−1 are prime powers, we have the inequality |ISO(Cqq )| >

|Aut(Fqq)|, and the equidistant code Cq

q is metrically nonrigid in a narrow sense. �Remark. Narrow-sense metric nonrigidity of a code with parameters (q, (q − 1)2, q − 1)q, where

q ≥ 5 and both q and q − 1 are prime powers, implies narrow-sense metric nonrigidity of a codewith parameters (q + 1, q2, q)q, where q ≥ 5 and both q and q − 1 are prime powers, which is aparticular case of the class of codes specified in Lemma 3 below, with μ = 1. We give a proof ofthe lemma, since it deals with metric nonrigidity of a wider class of codes. Note that this remarkdoes not apply to Proposition 1, where the code distance of the original and resulting codes is thesame.

Lemma 3. An optimal equidistant code Cnq with parameters

n =q2μ− 1

q − 1, M = q2μ, d = qμ, q ≥ 3,

where μ ≥ 1, is metrically nonrigid in a narrow sense.

Proof. Consider the set of isometries I : Cnq → Cn

q and the automorphism group of the space Fnq .

Let us show that the cardinality of the set of isometries is greater than the order of the automor-phism group of Fn

q , i.e., that|ISO(Cn

q )| > |Aut(Fnq )|. (3)

Then by Lemma 1 the code Cnq is metrically nonrigid in a narrow sense.

Note that

|AutFnq | = n! (q!)n =

(q2μ− 1

q − 1

)

! (q!)q2μ−1q−1 .

Since Cnq is an equidistant code, we have

|ISO(Cnq )| = M ! = (q2μ)!.


22 KOVALEVSKAYA

The following cases are possible.

Case 1. Let q = 3. Compare the numbers

(q!)q2μ−1q−1 = 6

9μ−12

and

q2μ(q2μ− 1) . . .

(q2μ− 1

q − 1+ 1

)

= 9μ(9μ − 1) . . .

(9μ− 1

2+ 2

)(9μ − 1

2+ 1

)

.

The product 9μ(9μ − 1) . . .(9μ− 1

2+ 2

)contains

9μ− 1

2factors, and we have

9μ− 1

2+ 2 ≥ 6;

therefore, 9μ . . .(9μ− 1

2+ 1

)> 6

9μ−12 . Hence we obtain

(9μ)! >

(9μ− 1

2

)

! · 69μ−1

2 ;

i.e., for q = 3 we have |ISO(Cn3 )| > |Aut(Fn

3 )|.Case 2. Let q = 4. Compare the numbers

(q!)q2μ−1q−1 = 24

16μ−13

and

q2μ(q2μ− 1) . . .

(q2μ− 1

q − 1+ 1

)

= 16μ(16μ − 1) . . .

(16μ − 1

3+ 1

)

= 16μ(16μ − 1) . . .

(

216μ − 1

3+ 1

)(

216μ − 1

3

)

. . .

(16μ − 1

3+ 1

)

.

Note that 216μ

3+ 1 > 6, and we have

16μ− 1

3+ 1 > 4 for any μ ≥ 1. Since each of the products

(16μ − 1) . . .

(

216μ − 1

3+ 1

) (

216μ − 1

3

)

. . .

(16μ − 1

3+ 1

)

contains16μ− 1

3factors, we obtain

(16μ − 1) . . .

(16μ − 1

3+ 1

)

> 616μ−1

3 · 416μ−1

3 = 2416μ−1

3 .

Hence,

(16μ)! >

(16μ − 1

3

)

! · 2416μ−1

3 ;

i.e., for q = 4 we have |ISO(Cn4 )| > |Aut(Fn

4 )|.Case 3. Let q ≥ 5. Then

(q2μ)! = q2μ . . .

(q2μ− 1

q − 1+ 1

)q2μ− 1

q − 1!.

Compare the numbers (q!)q2μ−1q−1 and q2μ . . .

(q2μ− 1

q − 1+ 1

). Note that

q2μ− 1

q − 1+ 1 > q, and the

product (q2μ− 1) . . .(q2μ− 1

q − 1+ 1

)contains

q − 2

q − 1(q2μ− 1) factors. Therefore,

(q2μ− 1) . . .

(q2μ− 1

q − 1+ 1

)

> qq−2q−1

(q2μ−1) = (qq−2)q2μ−1q−1 > (q!)

q2μ−1q−1

for q ≥ 5. Hence, (q2μ)! >q2μ− 1

q − 1! (q!)

q2μ−1q−1 ; i.e., for q ≥ 5 we have |ISO(Cn

q )| > |Aut(Fnq )|.



Thus, inequality (3) is valid for any q ≥ 3. Hence, not every isometry of the code is extendableto an automorphism F

nq , and therefore Cn

q is metrically nonrigid in a narrow sense. �The following lemma generalizes the example from Section 1. Zero columns in a reduced code

matrix make it possible to consider wider classes of codes. Furthermore, this lemma ensures metricnonrigidity for codes with the same parameters from some class of codes if the class contains atleast two nonequivalent codes (see Theorem 1 below).

Lemma 4. For any k ≥ 1 there exist at least k+1 nonequivalent equidistant codes with param-eters (n, [2n/d], d) provided that

√6k2 + 4nk − 4k < d ≤ n/2, d ≡ 0 (mod 2).

Proof. We consider two cases.

Case 1. Let k = 1. Consider the code C ={u0, . . . , u[ 2n

d]−1

}, where u0 = 0n and ui =

(1

d2 , 0(i−1) d

2 , 1d2 , 0n−(i+1) d

2); i.e., the code matrix of C is

⎡

⎢⎢⎢⎢⎢⎣

0 . . . 00 . . . . . . . . . . . . . . . 0 . . . 01 . . . 11 1 . . . 11 0 . . . 00 . . . . . . . . . 0 . . . 01 . . . 11 0 . . . 00 1 . . . 11 0 . . . 00 0 . . . 00 . . . 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . 11︸︷︷︸

d2

0 . . . 00︸︷︷︸d2

0 . . . 00︸︷︷︸d2

. . . . . . 1 . . . 11︸︷︷︸d2

0 . . . 0︸︷︷︸n−[ 2n

d] d2

⎤

⎥⎥⎥⎥⎥⎦

.

We construct a code D ={v0, . . . , v[ 2n

d]−1

}as follows:

v0 = 0n, vi =(1

d2 , 0(i−1) d

2 , 1d2 , 0n−(i+1) d

2)

for i <[2n

d

]− 1,

v[ 2nd]−1 =

(1

d2−1, 0

d2 , 1, 0

d2−1, 1, 0

d2−1, 1, . . . , 0

d2−1, 1

d2−[ 2n

d]+4, 0n−[ 2n

d] d2+[ 2n

d]−3);

i.e., the code matrix of D is of the form

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

0 . . . 00 . . . . . . . . . . . . . . . 0 . . . 01 . . . 11 1 . . . 11 0 . . . 00 . . . . . . . . . 0 . . . 01 . . . 11 0 . . . 00 1 . . . 11 0 . . . 00 . . . . . . 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . 11 0 . . . 00 0 . . . 00 . . . 1 . . . 11 0 . . . 00 0 . . . 01 . . . 10︸︷︷︸

d2

0 . . . 01︸︷︷︸d2

0 . . . 01︸︷︷︸d2

. . . 0 . . . 01︸︷︷︸d2

1 . . . 10 . . . 0︸︷︷︸d2

0 . . . 0︸︷︷︸n−[ 2n

d] d2

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.

Clearly, both codes C and D are equidistant.

If the codes C and D are equivalent, then dimΓ(C) = dimΓ(D). Compare the dimensions of Cand D; to this end, consider the corresponding code matrices. In the code matrix of C there are

n−[2n

d

]d

2zero columns, and in the code matrix of D there are precisely

n−[2n

d

]d

2+

([2n

d

]

− 3

)

zero columns. Note that for d ≤ n

2these quantities do not coincide. Therefore, dimΓ(C) =

dimΓ(D). Hence, there is no isometry of Fn that takes C to D, so the codes are not equivalent.


24 KOVALEVSKAYA

One should note that the code D exists ifd

2−

[2n

d

]+ 3 ≥ 0; otherwise, distances between

the codewords v[ 2nd]−1 and vi, where 1 ≤ i <

[2n

d

]− 1, do not equal d, and the code is no more

equidistant.

Case 2. Let k ≥ 2. We construct codes C0, . . . , Ck as follows:

C0 = {u0, . . . , u[ 2nd]−1}, where u0 = 0n, uj =

(1

d2 , 0(j−1) d

2 , 1d2 , 0n−(j+1) d

2),

Ci ={v0, . . . , v[ 2n

d]−2, v

i[ 2nd]−1

}, where v0 = 0n, vj =

(1

d2 , 0(j−1) d

2 , 1d2 , 0n−(j+1) d

2),

vi[ 2nd]−1

=(1

d2−i, 0

d2 , 1i, 0

d2−i, 1i, . . . , 0

d2−i, 1

d2−i[ 2n

d]+4i, 0n−[ 2n

d](d

2−i)−3i).

The codes Ci, 1 ≤ i ≤ k, exist ifd

2− k

[2n

d

]+ 3k ≥ 0; otherwise, they are not equidistant.

It is easily seen that such codes are equidistant. In the code matrix of C0 there are n −[2n

d

]d

2zero columns, and in the code matrix of Ci, i ≥ 1, there are precisely

n−[2n

d

]d

2+ i

([2n

d

]

− 3

)

zero columns. For d ≤ n

2these quantities do not coincide; hence, dimΓ(Ci) = dimΓ(Cj) for

0 ≤ i, j ≤ k, i = j. Therefore, there is no isometry of Fn that takes Ci to Cj for 0 ≤ i, j ≤ k, i = j,so the codes are not equivalent.

One should note thatd

2− k

[2n

d

]+ 3k ≥ 0 if and only if

d

2+ 3k > k

(2n

d− 1

)

,

and this inequality holds if d2 + 8dk − 4nk > 0. Hence, we obtain

d >√16k2 + 4nk − 4k.

For[2n

d

]= 2 there is a unique (up to equivalence) code with these parameters. The same holds

for[2n

d

]= 3. The first nonequivalent codes appear when

[2n

d

]= 4; therefore, d ≤ n

2. �

Theorem 1. Equidistant codes with parameters

(a)(q2μ− 1

q − 1, q2μ, qμ

)

qfor q ≥ 3, μ ≥ 1;

(b)(q, (q − 1)2, q − 1

)

qfor q ≥ 5, where both q and q − 1 are prime powers;

(c)(n, [2n/d], d

)for

√16 + 4n − 4 < d ≤ n/2, d ≡ 0 (mod 2),

are metrically nonrigid.

Proof. Metric nonrigidity of the equidistant codes with parameters(q2μ− 1

q − 1, q2μ, qμ

)

qand

(q, (q−1)2, q−1)q directly follows from Lemmas 1–3 and from the fact that a narrow-sense metricallynonrigid code is metrically nonrigid.

By Lemma 4, for √16 + 4n − 4 < d ≤ n/2, d ≡ 0 (mod 2),

there exist at least two nonequivalent codes, C and C ′, with parameters (n, [2n/d], d). Therefore,there is no isometry of Fn that takes C to C ′. Since both C and C ′ are equidistant, any map Ifrom the set of codewords of C to the set of codewords of C ′ is an isometry I : C → C ′; therefore,not every isometry of C is extendable to an isometry of the whole space Fn, so the equidistant codewith parameters (n, [2n/d], d) is metrically nonrigid. �



Note that a code corresponding to an affine resolvable design with parameters (n, qs, s, qμ, λ) isnot equidistant, since the distance between vectors of the same parallel class is 2qμ, whereas thedistance between vectors from different parallel classes is 2(qμ − μ).

Theorem 2. A code corresponding to an affine resolvable design with parameters (n, qs, s, qμ, λ),where

n = q2μ, q ≥ 3, s ≥ ncμq , cμq = log2 n/ log2(q!n),

is metrically nonrigid.

Proof. Let Cn be a code with parameters (n, qs, d) corresponding to an affine resolvable design Swith parameters (n, qs, s, qμ, λ) for some d.

Consider the set of isometries I : Cn → Cn and the automorphism group of Fn. Similarly toLemma 3, let us show that the cardinality of the set ISO(Cn) of isometries of Cn is greater thanthe order of the automorphism group of Fn, i.e.,

|ISO(Cn)| > |Aut(Fn)|.

Hence, using Lemma 1, we obtain that Cn is metrically nonrigid in a narrow sense, and is thereforemetrically nonrigid.

Note that|Aut(Fn)| = n! · 2n.

Since S is an affine resolvable design with s parallel classes, we have

|ISO(Cn)| = s! q!s.

By the condition, q ≥ 3. By supp(x) we denote the support of a vector x, i.e., the set ofcoordinates of x that are equal to 1: supp(x) = {i : xi = 1}. Consider the action of an arbitraryautomorphism (π, v) of Fn on a code corresponding to an affine resolvable design. Since the code isconstant-weight (each block contains precisely qμ elements) and since applying the permutation πto all vectors of the code does not change the weights of codewords, we conclude that for the codeto be fixed under the action of the automorphism (π, v), we need that adding the vector v to allvectors of the code also does not change the weights of the codewords. Therefore, the number ofcoordinates of v that are equal to 1 and belong to the support supp(x) of an arbitrary vector xof the code Cn must coincide with the number of coordinates of v that are equal to 1 and do notbelong to the support of x. Hence, the vector v must have even weight wt(v) = 2l for some l, andit must be such that for all vectors x in Cn we have

|{i : i ∈ v, i ∈ x}| = |{j : j ∈ v, j ∈ (1n − x)}|.

Since the design is affine resolvable, the set of blocks of the design can be partitioned into sdisjoint parallel classes. Consider an arbitrary parallel class, consisting of vectors x1, x2, x3, . . . , xq.

If wt(v) = 2l, then precisely l coordinates of v equal to 1 must belong to the support supp(x1)of x1, and they coincide with the unit coordinates of v that do not belong to the support supp(x2)of x2. The remaining l coordinates of v equal to 1 must in this case belong to supp(x2). However, bythe definition of a parallel class, the supports of the vectors x1, x2, and x3 are disjoint; moreover, allones of v are already counted, i.e., they belong to either supp(x1) or supp(x2), and therefore amongthe unit coordinates of v there is no coordinate that belongs also to supp(x3). Hence, the vectorsv and x3 are disjoint, and the equality wt(x3 + v) = qμ + 2l holds. Using the condition that theweight of all vectors of the code remains unchanged, we conclude that the only suitable translationvector is v = 0n. Therefore, there are at most n! automorphisms fixing the code. It remains todetermine when the condition s! q!s > n! holds.


26 KOVALEVSKAYA

1. For s ≥ n, this inequality is obviously valid.

2. Let s < n, s = n− x for some x. In this case the inequality

(n− x)! (q!)n−x > n!

holds if and only if

(q!)n−x > n . . . (n− x+ 1). (4)

The product n . . . (n−x+1) contains precisely x terms; therefore, if (q!)n−x ≥ nx, then (4) holds.In turn, the inequalities (q!)n−x ≥ nx and (q!)n ≥ (q!n)x hold if and only if x ≤ n log2(q!)/ log2(q!n).Hence we get

s ≥ n− n log2(q!)/ log2(q!n) = n log2 n/ log2(q!n).

Therefore, for s ≥ n log2 n/ log2(q!n) we have s! q!s > n!, and the code Cn corresponding to theaffine resolvable design is metrically nonrigid in a narrow sense. It remains to note that a narrow-sense metrically nonrigid code is metrically nonrigid. �

An equidistant code is said to be deadlock if no codeword can be added to it without changingthe code distance. An equidistant code with parameters (n,M, d)q is said to be maximal if thereare no equidistant codes with parameters (n,M + 1, d)q and (n,M, d+ 1)q. A binary Reed–Mullercode RM(r,m) of order r, 0 ≤ r ≤ m, of length 2m is defined as the set of vectors of length 2m

corresponding to all polynomials in m variables of degree at most r (see, e.g., [11]).

Proposition 2. Equidistant codes with parameters (n, q, 3)q , n ≥ 4, q ≥ 10, and (6, 6, 4)3, aswell as the first-order Reed–Muller code RM(1,m) and the punctured first-order Reed–Muller codeRM∗(1,m), m ≥ 2, are metrically nonrigid.

Proof. According to [10], there are two nonequivalent maximal equidistant codes with param-eters (n, q, 3)q for n ≥ 4 and q ≥ 10 and three nonequivalent deadlock equidistant codes withparameters (6, 6, 4)3. Hence, there exist isometric but not equidistant codes with fixed parameters,and this means that these codes are metrically nonrigid. Metric nonrigidity of the codes RM(1,m)and RM∗(1,m) follows from Lemma 1 and from the fact that a narrow-sense nonrigid code ismetrically nonrigid. �

Proposition 3. The Nordstrom–Robinson code, first-order Reed–Muller code RM(1,m), punc-tured first-order Reed–Muller code RM∗(1,m), and equidistant codes with parameters (n, [n/2], 4),n ≥ 17, (6, 4, 4)q , q = 3, 4, (5, 6, 4)3, and (6, 7, 4)3 are weakly metrically rigid.

The codes RM(1,m) and RM∗(1,m) and the Nordstrom–Robinson code (for a definition ofthis code, see, e.g., [11]) are weakly metrically rigid due to the fact that each of them is uniqueup to equivalence. The same holds for the equidistant codes with the parameters specified in theproposition, since they are unique according to [10].

Proposition 4. Equidistant codes with parameters (n, [2n/d], d), where

√16 + 4n − 4 < d ≤ n/2, d ≡ 0 (mod 2),

as well as those with parameters (n, q, 3)q, where n ≥ 4, q ≥ 10, and with parameters (6, 6, 4)3 areweakly metrically nonrigid.

These codes are weakly metrically nonrigid due to the fact that there exist isometric butnonequivalent codes with the given parameters (a construction of codes of the first class is given inLemma 4; existence of nonequivalent codes with the other parameters is proved in [10]).

Weak metric rigidity for other codes from Theorems 1 and 2 remains an open question, since wedo not know whether there exist nonequivalent codes with such parameters.



In conclusion, the author expresses her deep gratitude to F.I. Solov’eva for setting the problemand for permanent attention and support, and also to S.V. Avgustinovich and V.A. Zinoviev foruseful discussions and remarks.

REFERENCES

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3. Solov’eva, F.I., Avgustinovich, S.V., Honold, T., and Heise, W., On the Extendability of Code Isometries,J. Geom., 1998, vol. 61, no. 1–2, pp. 3–16.

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8. Semakov, N.V. and Zinoviev, V.A., Equidistant q-ary Codes with Maximal Distance and ResolvableBalanced Incomplete Block Designs, Probl. Peredachi Inf., 1968, vol. 4, no. 2, pp. 3–10 [Probl. Inf.Trans. (Engl. Transl.), 1968, vol. 4, no. 2, pp. 1–7].

9. Khachatrian, L.G., Equidistant Codes and Equidistant Permutation Arrays, Probl. Peredachi Inf., 1981,vol. 17, no. 4, pp. 116–119.

10. Bogdanova, G.T., Zinoviev, V.A., and Todorov, T.J., On the Construction of q-ary Equidistant Codes,Probl. Peredachi Inf., 2007, vol. 43, no. 4, pp. 13–36 [Probl. Inf. Trans. (Engl. Transl.), 2007, vol. 43,no. 4, pp. 280–302].

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On metric rigidity for some classes of codes