On unbalanced Boolean functions with best correlation immunity
Denis S. Krotov, Konstantin V. Vorob'ev (Sobolev Institute of, Mathematics, Novosibirsk, Russia)
TL;DR
This paper investigates the structure of unbalanced Boolean functions with maximum possible correlation immunity, establishing divisibility conditions, classifying certain equitable partitions, and characterizing related orthogonal arrays.
Contribution
It proves divisibility constraints for correlation-immunity bounds, classifies specific equitable partitions of the 12-cube, and characterizes related orthogonal arrays, advancing understanding of Boolean function structures.
Findings
CB is divisible by 3 for functions attaining the correlation-immunity bound
Exactly 2 equivalence classes of equitable partitions for a specific quotient matrix in 12-cube
Characterization of orthogonal arrays OA(1024,12,2,7) and OA(512,11,2,6)
Abstract
It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in variables cannot exceed ; moreover, it is if and only if the function corresponds to an equitable -partition of the -cube with an eigenvalue of the quotient matrix. The known series of such functions have proportion , , or of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio of the number of ones and zeros, then is divisible by . In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly equivalence classes of the equitable partitions of the -cube with quotient matrix and classes, with .…
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On unbalanced Boolean functions with
best correlation immunity††thanks: This work was funded by the Russian Science Foundation under grant 18-11-00136.
Denis S. Krotov Konstantin V. Vorob’ev
Sobolev Institute of Mathematics
Novosibirsk 630090, Russia
{krotov,vorobev}@math.nsc.ru
Abstract
It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in variables cannot exceed ; moreover, it is if and only if the function corresponds to an equitable -partition of the -cube with an eigenvalue of the quotient matrix. The known series of such functions have proportion , , or of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio of the number of ones and zeros, then is divisible by . In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices.
We also establish that there are exactly equivalence classes of the equitable partitions of the -cube with quotient matrix and classes, with . These parameters correspond to the Boolean functions in variables with correlation immunity and proportion and , respectively (the case remains unsolved). This also implies the characterization of the orthogonal arrays OA and OA.
1 Introduction
We study unbalanced Boolean functions with the maximum possible order of correlation immunity. A function is called unbalanced if the number of its ones is different from [math], , and . It is called -th order correlation immune if the number of ones (equivalently, zeros) is statistically independent on the values of any arguments. Fon-Der-Flaass [5] proved that the correlation-immunity order of an unbalanced Boolean function in variables cannot exceed ; moreover, any unbalanced Boolean function of correlation-immunity order corresponds to an equitable -partition of the -cube with quotient matrix , where and (a formal definition can be found in Section 2; here, it is essential that the number of ones of relates to the number of zeros as ). Nowadays, there are three known families of quotient matrices corresponding to such functions: , (found in [23]), (found in [7]). For each of the matrices , , and , a function is unique up to equivalence. Kirienko [13] found that there are exactly two inequivalent unbalanced Boolean functions in variables attaining the bound on the order of correlation immunity (the corresponding quotient matrix is ). Fon-Der-Flaass [7] started the investigation of the equitable partitions of attaining the correlation-immunity bound. It was shown that equitable partitions with quotient matrix do not exist, while equitable partitions with quotient matrix were built (see the construction in Section 5). These results were also important from the framework of the study of parameters of equitable -partitions of the -cube: they closed the smallest open cases remaining after the general paper [6]. After that, all quotient matrices of equitable -partitions of the -cube were characterized for any smaller than . For , the remaining questionable matrices were , , , , , and it is notable that all these parameters correspond to unbalanced Boolean functions with extreme order of correlation immunity, .
In the present work, we prove a new property of the equitable partitions that meet the correlation-immunity bound with equality. In particular, our results imply the nonexistence of an equitable partition with quotient matrix or , as well as any Boolean function with correlation immunity and proportion between the number of ones and the number of zeros , , or any such that is not divisible by . Besides that, we provide a characterization of all inequivalent equitable partitions with the quotient matrices and .
From the theoretical point of view, studying Boolean functions lying on the correlation-immunity bound with different proportions of the number of ones and zeros is the most intriguing part of our research. On the other hand, the functions of correlation-immunity order with ones are of special interest because of the following two connections, and our classification related with the quotient matrix makes a contribution to their study.
The first connection is with -resilient functions. A function is called -resilient if for every from the function
[TABLE]
is correlation immune of order with ones. The resilient functions are important for applications in cryptography, see e.g. [4]. If , then [8]. If and , then the functions belong to the class of functions we study and correspond to the equitable partitions of with quotient matrix , .
The second connection is with orthogonal arrays. An orthogonal array OA (we consider only the binary orthogonal arrays) is a multiset of vertices on the -cube such that the number of its elements with prescribed values in any positions does not depend on those values, see e.g. [9]. (Often, the elements of an orthogonal array are considered as being arranged as the rows or the columns of an or array, respectively). An orthogonal array is simple if it is an ordinary set, without multiplicities more than one. It is straightforward that the simple OA are in one-to-one correspondence with the Boolean functions of correlation-immunity order with ones (actually, the set of ones of such function forms the corresponding OA). A result of Bierbrauer [1, Theorem 1] says for OA that
[TABLE]
moreover, for a non-simple array the inequality becomes strict, which is straightforward from the proof, see [1, p. 181, line 4] (note that for the simple binary orthogonal arrays, the bound was proved earlier by Friedman [8, Theorem 2.1]). The arrays OA lie on this bound; hence, they are simple and, as follows from the results of [5], correspond to the equitable partitions with quotient matrix , . In particular, the results of our classification imply that there are exactly inequivalent OA and exactly inequivalent OA (such arrays are obtained from OA by shortening). The classification of orthogonal arrays with given parameters is a problem that attracts attention of many researchers, see the recent works [2], [3], and the bibliography there. We note that the preceding computational results were successful for smaller and than in our case. Futher discussion on the quotient matrix , related equitable partitions, orthogonal arrays, and Boolean functions can be found in [15].
With the other parameters considered in our paper, the situation is different. The Fon-Der-Flaass bound was generalized to the binary orthogonal arrays by Khalyavin [12], who proved that any OA with satisfies . However, this does not mean that any array lying on this bound is simple (e.g., there is a non-simple OA [24]). The classification of non-simple orthogonal arrays that meet the Fon-Der-Flaass–Khalyavin bound is a separate problem, which is not considered in the current research.
The introductory part of the paper continues with definitions and basic facts (Section 2) and Section 3, where we describe the computer tools used for the classification results. The main theoretical results of the paper are proved in Section 4. Theorem 5 states (in an equivalent formulation) that if the correlation-immunity order of an unbalanced Boolean function lies on the Fon-Der-Flaass bound, then the number of ones of the derivative of in any basic direction , , does not depend on the direction. As a consequence, we have a new necessary condition on the existence of such functions and corresponding equitable -partitions (Corollary 6). Section 5 contains the characterization of inequivalent equitable -partitions of the -cube with quotient matrix , based on the combination of theoretical and computational results, and a description of the original Fon-Der-Flaass construction [7] of such partitions, including the representation via the Fourier transform. In Section 6, we describe the computational classification of the equitable partitions of the -cube with quotient matrix . The list of the all inequivalent partitions is given in the appendix. As we mentioned above, the last partitions correspond to the order- correlation immune Boolean functions in variables with ones, and to the orthogonal arrays OA. In Section 7, we briefly discuss equitable partitions with quotient matrix and the connection of such partitions with orthogonal arrays OA.
2 Definitions and basic facts
Let be an undirected graph. A partition of the vertex set into cells is called an equitable partition (equitable -partition) with quotient matrix if for all any vertex of has exactly neighbors in . Two partitions and of are equivalent if there is a graph automorphism such that for every from .
The -cube (also known as the Hamming graph ) is a graph whose vertices are the words of length over the alphabet , also treated as vectors over the binary field GF(). Two vertices are adjacent if and only if they differ in exactly one coordinate position, which is referred to as the direction of the corresponding edge. The Hamming distance between vertices and is the number of coordinates in which they differ. The weight of a word is the number of ones in it. By , we denote the ordinary inner product of vectors: . For two vertices , , we will write if for all from to . We denote by the word with all zeros except one in the -th position; by and , the all-zero and all-one words, respectively. For , the set is a -face of , where is the dimension of the face.
Any equitable -partition of satisfies the following necessary conditions on the coefficients of its quotient matrix [6], [5]:
- (a)
, , , are nonnegative integers such that , , ; 2. (b)
is a power of ; 3. (c)
if , then .
Condition (c) is a special case of the bound on the order of correlation immunity of an unbalanced Boolean function [5]. We say that an equitable -partition with quotient matrix attains the correlation-immunity bound if and (equivalently, ). Besides (a)–(c), the only matrix for which the nonexistence of the corresponding equitable -partitions of was established, before the current research, was (usually, we also agree that , because this can always be reached by choosing the order of cells), see [7]. In Section 4, we will prove a new necessary condition, which rejects the matrix as well as an infinite number of other matrices satisfying (a)–(c).
Some of our results are formulated in terms of real-valued functions defined on the vertices of the -cube. Two such functions are equivalent if there is a permutation of coordinate positions and a vector such that for all . The norm of a function is .
Given an eigenvalue of the adjacency matrix of a graph , a function is called an eigenfunction or a -eigenfunction of if it is not constantly zero and for every
[TABLE]
Note that the tuple of values of a -eigenfunction is essentially an eigenvector of the adjacency matrix of corresponding to the eigenvalue .
It is well known and easy to check that the eigenspectrum of is and the set of functions is an orthogonal basis of the -eigenspace of for . Therefore, for a function defined on , the following identity holds
[TABLE]
where
[TABLE]
is a Fourier coefficient, . By the weight of the coefficient , we will understand the weight of . The next properties of the basis follow instantly from its definition.
Proposition 1**.**
For the following equalities hold:
- (i)
,
- (ii)
.
We will need the following well-known properties of the basis functions.
Proposition 2**.**
- (i)
For every , every of weight , and every -face , it holds
[TABLE]
- (ii)
(see, e.g., [17, Ch. 5, Lemma 2])* for every and from , it holds*
[TABLE]
Proof.
(i) Let for some words of weight and . Since , there is some coordinate position where and both have . Thus, for every , we have .
(ii) By the definition of a Fourier coefficient, equals if and [math] otherwise. Thus, the left side of (3) is equal to if and zero otherwise. In the right side, we also have if . In the remaining case , we have [math] by arguments similar to (i). ∎
For a given set of elements, a -covering, , is a set of -subsets of such that for every -subset of there exists from such that . The following facts are trivial and well known, see e.g. [22].
Proposition 3**.**
If be a -covering of a set of size , then
- (i)
;**
- (ii)
for every , the set is a -covering of .
Given an equitable -partition of with quotient matrix , by its associated function we will understand the function defined as follows:
[TABLE]
Lemma 4** ([5, 7]).**
Let be an equitable -partition of with quotient matrix and associated function . Then the following identities take place:
[TABLE]
Proof.
Counting the values of over the neighbours of a given vertex, we find that is an -eigenfunction of . Thus, all its nonzero Fourier coefficients have weight . By the definition of the associated function, we know that . Therefore,
[TABLE]
After removing parentheses and using Proposition 1, we obtain the remaining equalities. ∎
The kernel of an equitable -partition is the set
[TABLE]
of all periods of the cells or, equivalently, of the associated function .
3 Computational tools
Exact covering.
The approaches we apply for enumerating equitable partitions of (the approaches are completely different for the quotient matrices and ) include solving instances of the exact covering problem. In general, the exact covering problem can be formulated as follows. Given elements , …, , natural numbers , …, , and a collection , …, of subsets of the set , …, , find a subcollection of such that each element is contained in exactly sets from . Most mathematical packages include methods for finding an exact cover in the case , which is solved much effectively in practice than the general problem. However, our approaches need finding exact covers with different multiplicities. We exploited the libexact package [11], which can be used in c/c++ programs.
Isomorphism.
To find the number of equivalence classes of -partitions of the vertices of from a considered class, or any intermediate objects, we use the standard technique described in [10, Sect. 3.3]. Namely, sets of vertices of are represented by graphs in such a manner that two objects are equivalent if and only if the corresponding graphs are isomorphic. A famous package to work with the graph isomorphism is nauty [18]. The same approach allows to find the automorphism group of any object we deal with.
Double counting.
The following nice approach, described in [10, Sect. 10.2], allows to partially validate the results of the exhaustive search. Assume that we have finished the classification of some objects and have found a representative of every equivalence class. Knowing the order of the automorphism group of each representative, we can calculate the total number of different objects. If this number does not coincide with the number of objects found by the exhaustive search, then the search was erroneous. This approach catches many kinds of systematic and random mistakes, but only works if the result of the search is not empty. We checked the results of every step of our classification by this double-counting method.
4 New necessary condition
In this section we provide a new necessary condition of the existence of equitable -partitions of attaining the bound [5] on correlation immunity. Given an equitable partition of a -cube, we will say that an edge of the graph is composite if it is incident to vertices from different cells of the partition.
Theorem 5**.**
Let be an equitable partition of with quotient matrix , , attaining the correlation-immunity bound, i.e., . Let be the function associated to this partition. The following statements are true:
- (i)
the value does not depend on ;
- (ii)
the number of composite edges of direction does not depend on .
Proof.
(i) Since our partition attains the bound on correlation immunity, we have . By Lemma 4, we know that if , and for every , the following equality holds:
[TABLE]
Take some . Consider the square of the norm of the subfunction corresponding to , where . Our goal is to show that this norm does not depend on the choice of .
[TABLE]
We state that
[TABLE]
Indeed, for every nonzero term , each of the words , , has exactly zeros and the positions of the zeros do not intersect for , , and (which follows from ). Therefore, every nonzero summand in the right side occurs exactly in one sum in the left side of the equality. This observation proves the last equality and the first claim of the theorem.
(ii) Let us count the number of composite edges of an arbitrary direction . Clearly, this value equals
[TABLE]
Using the Fourier transform, we have that
[TABLE]
After removing parentheses, dividing by , and changing the order of summing, we have
[TABLE]
By claim (i), the proof is done. ∎
Corollary 6**.**
If there exists an equitable partition of with quotient matrix attaining the correlation-immunity bound, , then either or is divisible by .
Proof.
Let be an equitable -partition of with quotient matrix , , attaining the correlation immunity bound. From the definition of an equitable partition, we see that there are vertices in . Consequently, there are exactly composite edges in the graph. By Theorem 5 we conclude that
[TABLE]
Since our partition attains the bound on correlation immunity, we have . The degree of the -cube equals ; so, we have . Substituting this expression to the number of edges, we prove the required statement. ∎
Corollary 6 implies the nonexistence of an infinite sequence of putative parameters of equitable -partition of for which this question was open before. In particular, it gives an alternative proof of the nonexistence of equitable -partitions of with quotient matrix , which was shown in [7], and the nonexistence of -partitions of with quotient matrices and :
Corollary 7** (example).**
There are no equitable -partitions of with quotient matrices () and ().
5 The equitable partitions with quotient matrix
In this Section we characterize all inequivalent -partitions of with quotient matrix .
5.1 General properties
Let be an equitable -partition with quotient matrix . By direct counting, we have and . Let be the associated function:
[TABLE]
By Lemma 4, we know that is an eigenfunction corresponding to the eigenvalue and all its nonzero Fourier coefficients have weight . Therefore, Proposition 2(i) guarantees that the sum of values of over any -face equals [math]. Consequently, any -face contains exactly vertices from and vertices from . Proposition 2(ii) gives us the identity
[TABLE]
In the right side of the equality we have the sum of values of over some -face of . This means that . In particular, is integer.
Let us take an arbitrary of weight and use Proposition 2(ii) one more time:
[TABLE]
Since the value from the right side of the equation belongs to , the sum is odd. For a given of weight , there is at least one of weight such that is odd. In other words, the set of quadruples of zero coordinates of the weight- vertices for which is odd forms a -covering . Our next goal is to describe the set of possible values can take.
Applying Lemma 4 to our function, we have
[TABLE]
Suppose there is such that . Without loss of generality we take
[TABLE]
By Theorem 5 we have
[TABLE]
By Proposition 3(ii), the elements of our covering containing the -th coordinate position form a -covering by odd values of of the set . Since the sum of squares equals and , we conclude that the size of this covering is not bigger than . By Proposition 3(i), it must be at least , and we get a contradiction.
The arguments above prove the following statement.
Lemma 8**.**
Let be the associated function of an equitable -partition of with quotient matrix . Then for every .
5.2 Configurations of overcovered triples
As follows from Lemma 8 above, for every triple of different coordinates the number of nonzeros of such that is odd. Since the nonzero has exactly zero coordinates (the quadruple of these coordinates will be referred to as block), we have a covering of all triples by blocks, a covering. Consider the multiset of all triples where the multiplicity of a triple is the number of blocks covering this triple (we know this number is odd). Reducing the multiplicities by , we get a multiset with the coefficients equal to the number of “overcovering” of the corresponding triple. All these coefficients are even, and hence we can divide them by , obtaining a multicet . The elements of will be called bitriples (naturally, one bitriple in corresponds to two triples in ). Taking into account the multiplicities, we have exactly bitriples. Indeed, blocks cover triples in total; each of -subsets of the set of coordinates is covered, plus each of bitriples is covered twice.
Lemma 9**.**
Every coordinate belongs to exactly bitriples.
Proof.
Every coordinate belongs to blocks, which cover triples with given coordinate, taking into account the multiplicities. The number of different such triples is . So, we have bitriples (recall that each of bitriples corresponds to two overcoverings, by the definition). ∎
Lemma 10**.**
(i)* Every two different coordinates belong to an odd number of blocks, (ii) at least .*
Proof.
Assume that the -st and -nd coordinates meet in exactly blocks. We know that the number of all blocks is ; exactly of them contain the first coordinate; exactly of them contain the first and the second coordinates. Hence, exactly blocks contain the first coordinate and do not contain the second. Similarly, exactly blocks contain the second coordinate and do not contain the first.
The sum of all values of is or (the value of in ).
(a) The value of at is also or ; therefore, among the nonzeros with in the first coordinate, either , or , or values and , , or values , respectively (for example, if and , then among nonzeros with in the first coordinate, should have the value and the value , for the sum change by during the sign inverse, which corresponds to the translation of the partition by the vector ).
(b) The value of at is also or , therefore, among the nonzeros with in the second coordinate, either , or , or values and , , or values , respectively.
(c) The value of at is also or , therefore, among the nonzeros with different values in the first and the second coordinate, either , or , or values and , , or values , respectively.
In the arguments (a), (b), (c), every nonzero occurs twice or does not occur at all (if it starts with ). Indeed, if we denote by the number of nonzeros such that , , and , then we get , , . Hence, is even. On the other hand, . It follows that is even and is odd.
(ii) follows from covering of all triples that include the given pair. ∎
Corollary 11**.**
Each two different coordinates belong to an even number of bitriples, [math], , or .
Proof.
Without loss of generality, consider the first two coordinates. every -block containing them covers exactly two triples they belong to. So, the number of such -blocks is the half of the number of different triples of form , , plus the number, say , of bitriples of such form. That is, . By Lemma 10, this number is odd. Hence, is even. ∎
Our next goal is to describe possible bitriple systems up to equivalence. We first assume that there is at least one bitriple of multiplicity .
Lemma 12**.**
*If there is a bitriple of multiplicity , then it belongs to the collection of bitriples , up to a coordinate permutation. *
Proof.
Without loss of generality assume that we have a bitriple of multiplicity . By Corollary 11, there is another bitriple with and . Without loss of generality it is . By Corollary 11, there is another bitriple with and . It cannot be , because in that case any choice of the forth bitriple with contradicts Corollary 11. So, it is , without loss of generality (we did not use before). By Corollary 11, the fourth element with is .
Again by Corollary 11 and since the multiplicity of is , there is another bitriple with and . If it is , then we have bitriples , , with , and the fourth bitriple with contradicts Corollary 11. A similar argument rejects (with respect to ). So, without loss of generality, we have .
Now, the fourth bitriple with must be ; the fourth bitriple with must be ; the fourth bitriple with must be . ∎
Lemma 13**.**
If there is a bitriple of multiplicity , then the multiset of bitriples is one of the following, up to a coordinate permutation:
[TABLE]
Proof.
By Lemma 12, we have the first bitriples. If there is another, -th bitriple of multiplicity , then by the same lemma we have (10). If there is no -th bitriple of multiplicity , then the remaining bitriples have multiplicity or , and a simple exhaust search results in (10)–(10). ∎
If there is no bitriple of multiplicity , then the multiplicities of bitriples are or . In this case, we can again divide them by , which results in a multiset of triples, call them bibitriples. Every coordinate is covered by exactly bibitriples. So, if there is no bibitriples of multiplicity , then the bibitriples form a - design. If there is exactly one bibitriple of multiplicity , the remaining form a - design. If there is exactly two bibitriples of multiplicity , the remaining form a - design. The remaining case is bibitriples of multiplicity . The number of - designs is known for , see http://oeis.org/A110100. In particular, up to permutation of the coordinates, we have , , and solutions, respectively.
Finally, we know that the multiset of bibitriples is one of equivalence classes.
5.3 Coverings by -ples
For each multiset of bitriples, we can find all possible systems of quadruples such that every triple is included times, where is its multiplicity in the multiset of bitriples. To do this, we have to solve the corresponding instance of the exact cover problem. This can be done in seconds on a modern computer (we used the libexact [11] package with c++). The result is as follows.
Proposition 14**.**
There are exactly equivalence classes of coverings such that the overcovered triples correspond to one of the equivalence classes of bitriples mentioned above. Only of equivalence classes of bitriples can be realized in this way; namely, (10) ( inequivalent coverings found), (10) ( covering), (10) ( coverings),
[TABLE]
(* covering*), and
[TABLE]
(* coverings*).
5.4 Finding signs of the Fourier coefficients
So, we have got candidates for the set of nonzeros. To find the Fourier coefficient in each nonzero, we will exploit equations (5), (6). In particular, for , we have
[TABLE]
where denotes lexicographic preceding. The last equation immediately gives a necessary condition on the set of nonzeros (indeed, for every nonzero , we have ; so, both parts of (12) do not depend on the sign of ). This condition rejects of coverings, as shown in the following computational proposition.
Proposition 15** (computational results).**
Among the coverings found in Proposition 14, exactly coverings can correspond to the nonzeros of a -valued function satisfying (12). All these coverings correspond to the set (10) of bitriples.
Now assume that is the set of nonzeros of and that the function defines the sign of in each nonzero:
[TABLE]
We will show that the values of satisfy a system of linear equations over , one equations for each .
Consider any zero of different from , i.e., , . By (12), the number of pairs of elements from such that is even (the pairs are unordered; so, we can always assume ). Denote this number by . From (12) we see that for pairs we have and for the rest pairs . It follows that the number of s among all such s and s has the same parity as . Let us write this fact as an equation.
[TABLE]
Next, consider an arbitrary nonzero . For simplicity assume that . From (12) we see that is odd, and we find from (12) that the number of s among all considered and is if and if . We derive the following identity.
[TABLE]
We see that the values of satisfy the system of equations (14), (15) over the finite field GF() of order (some of the equations (14) are trivial, ; so, the actual system to solve has less than equations). This system can be solved for all of the remaining candidates for .
Proposition 16** (computational results).**
Among the sets considered in Proposition 15, the system of equations (14), (15) is consistent for exactly sets. In each of these cases, the rank of the system is ; so, the number of solutions is .
It remains, among the solutions in each of cases, to choose the functions that correspond to the Fourier transform of -valued functions. It is doable in a reasonable time; however, the following observation reduces the number of calculations even more.
Lemma 17**.**
For each from to , define the coordinate function by the identity , where .
- (i)
If some satisfies all equations (14), (15) then does.
- (ii)
Moreover, if , see (13), is the Fourier transform of a -valued function, then adding to does not change this property.
Proof.
(i) It is easy to see that the number of s and s involved in (14) that have in the -th position is even. Indeed, if , then for every pair under the sum. If , then , but the number of the pairs involved in the sum is even. Hence, adding does not change the parity of the left side of (14).
The similar argument works for (15) with the only difference that is even, which is compensated by involving in the left side.
(ii) It is straightforward from the definition of the Fourier transform that the sum corresponds to the translation , where has in the -th position and [math] in the others. ∎
So, the affine space of the all solutions can be partitioned into the cosets of the span (the span has dimension in one of the remaining cases and dimension in the other), and it is sufficient to test one representative from every coset. Finally we have found admissible representatives in one of the cases and in the other. It occurs that in each of two cases, the equitable partitions found are equivalent.
Theorem 18**.**
There are exactly inequivalent equitable partitions of with quotient matrix . Each of them has the automorphism group of order ; the sizes of orbits under the action of the automorphism group are , , for the smallest cell (in the notation ) and , for the largest cell. One partition is coordinate transitive (that is, the coordinates form one orbit under the action of the automorphism group) and the size of its kernel is . The other partition has two coordinate orbits of size and the kernel of size .
5.5 Fon-Der-Flaass construction
In this section, we define the equitable partitions constructed by Fon-Der-Flaass [7] and describe the corresponding Fourier transforms.
First, we color the vertices of into three colors as follows (symbol can be replaced by each of [math] and ; so, a word like represent a set from vertices, which is referred to as a -face).
[TABLE]
Next, color as .
It remains to separate gray vertices into white and black. For every -face from the twelve -faces above, with s in the -th and -th position, we color such that with respect to the parity (by black/white or white/black; so we have the choice for each -face). In such a way, we obtain different black/white colorings of the vertices of ; the corresponding vertex partitions are equitable with quotient matrix [7].
By Proposition 2(ii) the Fourier coefficient at (e.g., is proportional (with ) to the sum of over the corresponding -face (e.g., respectively, ). So, the coefficients are straightforward to find. We omit technical details and describe the possibilities corresponding to the partitions constructed above. The nonzeros of one possible Fourier transform, with the corresponding signs, are the following:
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\bar{u}\\[-2.84526pt] \bar{v}\end{array}\Big{]}: \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}1\,111\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}1\,111\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}1\boldsymbol{0}\,111\\[-2.84526pt] \boldsymbol{0}1\boldsymbol{0}\,111\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}\boldsymbol{0}\,111\\[-2.84526pt] 1\boldsymbol{0}\boldsymbol{0}\,111\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}111\,\boldsymbol{0}\boldsymbol{0}1\\[-2.84526pt] 111\,\boldsymbol{0}\boldsymbol{0}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}111\,\boldsymbol{0}1\boldsymbol{0}\\[-2.84526pt] 111\,\boldsymbol{0}1\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}111\,1\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] 111\,1\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{+}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,\boldsymbol{0}11\\[-2.84526pt] \boldsymbol{0}11\,\boldsymbol{0}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,1\boldsymbol{0}1\\[-2.84526pt] \boldsymbol{0}11\,1\boldsymbol{0}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,11\boldsymbol{0}\\[-2.84526pt] \boldsymbol{0}11\,11\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,\boldsymbol{0}11\\[-2.84526pt] 1\boldsymbol{0}1\,\boldsymbol{0}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,1\boldsymbol{0}1\\[-2.84526pt] 1\boldsymbol{0}1\,1\boldsymbol{0}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,11\boldsymbol{0}\\[-2.84526pt] 1\boldsymbol{0}1\,11\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,\boldsymbol{0}11\\[-2.84526pt] 11\boldsymbol{0}\,\boldsymbol{0}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,1\boldsymbol{0}1\\[-2.84526pt] 11\boldsymbol{0}\,1\boldsymbol{0}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,11\boldsymbol{0}\\[-2.84526pt] 11\boldsymbol{0}\,11\boldsymbol{0}\end{array}\Big{]}\!{+}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,11\boldsymbol{0}\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,11\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,11\boldsymbol{1}\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,11\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,11\boldsymbol{1}\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,11\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,11\boldsymbol{0}\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,11\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,11\boldsymbol{1}\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,11\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,11\boldsymbol{0}\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,11\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,11\boldsymbol{0}\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,11\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,11\boldsymbol{1}\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,11\boldsymbol{0}\end{array}\Big{]}\!{+}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,1\boldsymbol{0}1\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,1\boldsymbol{1}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,1\boldsymbol{1}1\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,1\boldsymbol{0}1\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,1\boldsymbol{1}1\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,1\boldsymbol{0}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,1\boldsymbol{0}1\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,1\boldsymbol{1}1\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,1\boldsymbol{1}1\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,1\boldsymbol{0}1\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,1\boldsymbol{0}1\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,1\boldsymbol{1}1\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,1\boldsymbol{0}1\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,1\boldsymbol{1}1\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,1\boldsymbol{1}1\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,1\boldsymbol{0}1\end{array}\Big{]}\!{+}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,\boldsymbol{0}11\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,\boldsymbol{1}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,\boldsymbol{1}11\\[-2.84526pt] \boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,\boldsymbol{0}11\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,\boldsymbol{1}11\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,\boldsymbol{0}11\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,\boldsymbol{0}11\\[-2.84526pt] \boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,\boldsymbol{1}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\,\boldsymbol{1}11\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\,\boldsymbol{0}11\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\,\boldsymbol{0}11\\[-2.84526pt] \boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\,\boldsymbol{1}11\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\,\boldsymbol{0}11\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\,\boldsymbol{1}11\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\,\boldsymbol{1}11\\[-2.84526pt] \boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\,\boldsymbol{0}11\end{array}\Big{]}\!{+}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] 11\boldsymbol{1}\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{1}\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] 11\boldsymbol{0}\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{1}\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] 11\boldsymbol{0}\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] 11\boldsymbol{1}\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{1}\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] 11\boldsymbol{0}\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] 11\boldsymbol{1}\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{0}\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] 11\boldsymbol{1}\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}11\boldsymbol{1}\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] 11\boldsymbol{0}\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{-}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] 1\boldsymbol{1}1\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{1}1\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] 1\boldsymbol{0}1\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{1}1\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] 1\boldsymbol{0}1\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] 1\boldsymbol{1}1\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{1}1\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] 1\boldsymbol{0}1\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] 1\boldsymbol{1}1\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{0}1\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] 1\boldsymbol{1}1\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}1\boldsymbol{1}1\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] 1\boldsymbol{0}1\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{-}
\displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] \boldsymbol{1}11\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}11\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] \boldsymbol{0}11\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}11\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] \boldsymbol{0}11\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] \boldsymbol{1}11\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{+} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}11\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{0}\\[-2.84526pt] \boldsymbol{0}11\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,\boldsymbol{1}\boldsymbol{0}\boldsymbol{1}\\[-2.84526pt] \boldsymbol{1}11\,\boldsymbol{0}\boldsymbol{1}\boldsymbol{0}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{0}11\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{0}\\[-2.84526pt] \boldsymbol{1}11\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{1}\end{array}\Big{]}\!{-} \displaystyle\Big{[}\begin{array}[]{@{}c@{}}\boldsymbol{1}11\,\boldsymbol{1}\boldsymbol{1}\boldsymbol{1}\\[-2.84526pt] \boldsymbol{0}11\,\boldsymbol{0}\boldsymbol{0}\boldsymbol{0}\end{array}\Big{]}\!{-}.
In each of the last six groups, all the signs can be inversed. Additionally, in each of the last six groups, one can apply the coordinate permutation to all nonzeros. The last transformation, applied to one group, switches between the two equivalence classes of the equitable partitions.
6 and related structures: classification
The equitable partitions of the -cube with quotient matrix (or, equivalently, the orthogonal arrays OA, as was mentioned in the introduction) can be classified utilizing rather straightforward approach, a local exhaustive search, using the exact-covering software. Let be the quotient matrix . We say that the pair of disjoint sets , of vertices is an -local (equitable) partition if are the all words of weight at most and the neighborhood of every vertex of weight less than satisfy the local condition from the definition of the equitable partition.
So, there are exactly two [math]-local partitions, and . For each of them, there is only one -local partition, up to isomorphism.
Proposition 19**.**
Up to isomorphism, there are exactly two-local partitions with , and exactly , with .
Proof.
Let . In this case, all weight- words are in . Consider the graph on the weight- words, where two vertices are adjacent if in the -cube they are adjacent to a common weight- word from . So, the weight- words from are in one-to-one correspondence with the edges of (indeed, a weight- word has exactly weight- neighbors). Next, we see that is a cubic graph (indeed, every weight- word is in and hence has exactly neighbors from ; one of them is , the other correspond to edges of ). The number of unlabelled connected cubic graphs on , , , and vertices is , , , , respectively, see http://oeis.org/A002851. So, the number of connected and disconnected cubic graphs on vertices is .
Let . Without loss of generality, all weight- words with in the first coordinates are assumed to be in , the other in . The last four words have no neighbors in ; so, any weight- word in has two weight- neighbors in and can be considered as an edge of some graph on vertices (weight- words of ). From the quotient matrix, we see that is regular of degree ; so, its complement is cubic. There are disconnected and connected cubic graphs of order . ∎
The search of the -local partitions was done by solving instances of the exact covering problem. We fix some -local partition and consider the weight- words in as the “points”. To each “point” , we assign the multiplicity , where is the number of its weight- neighbors from . To each weight- word that has no neighbors from , we assign a “set” of “points”, namely the weight- neighbors of . With the chosen “points”, their multiplicities, and the “sets”, we have an instance of the exact-covering problem. Straightforwardly from the definitions, we have the following one-to-one correspondence.
Proposition 20**.**
Given a -local partition , the -local partitions such that and are in one-to-one correspondence with the solutions of . Namely,
In such a way, for each of non-isomorphic -local partitions, using libexact, we found all -local continuations. After the isomorph rejection, we found all non-isomorphic -local partitions. The same approach allows to proceed the next step in finding the -local partitions. The results are checked using the double-counting approach (see Section 3).
Proposition 21** (computational results).**
The number of non-isomorphic -local partitions with and is and , respectively. For -local partitions, the number is and , respectively.
The remaining part of the classification is based on the fact that the sum of the values of the -valued eigenfunction corresponding to a putative equitable partition (with considered parameters) over any -face is zero. Using this condition, one can uniquely reconstruct an eigenfunction by its values on the words of weight at most (actually, it is sufficient to know the values on the weight- words, see [25, Theorem 3]). It occurs that every -local partition continues to a complete equitable partition (we have no theoretical proof of this fact).
Theorem 22** (computational results).**
There are exactly equivalence classes of equitable partitions of the -cube with quotient matrix . In one of them, is a linear (or affine) subspace of the -cube; two are “full-rank”, i.e., the affine span of is the whole -cube; the other are “semilinear”, that is, the affine span of consists of a half of the vertices of the -cube. See the appendix for the list of representatives.
Remark 23*.*
For the classification, it is sufficient to consider only the local partitions that meet , or only the local partitions that meet . However, as the both ways were successful, we described in Propositions 19 and 21 the intermediate results for each of them.
Remark 24*.*
The local search algorithm described in this section can be applied for finding equitable partitions with different parameters (in different graphs). However, we failed in the classification of the equitable partition of the -cube with quotient matrices and using the same approach. The corresponding instances of the exact-covering problem occur to be too large to solve with known tools.
The equitable partitions considered in the current section are related with several classes of combinatorial configurations. The following lemma summarizes several known results about such relations.
Lemma 25**.**
The objects from the following classes are in one-to-one correspondence:
- (I)
the equitable partitions of the -cube with quotient matrix \left(\begin{array}[]{cc}0&n\\ c&n-c\end{array}\right), ;
- (II)
the orthogonal arrays OA, where and \displaystyle N=2^{n}\Big{(}1-\frac{n}{2(t+1)}\Big{)} (so, the parameters attain the bound (2));
- (III)
the orthogonal arrays OA;**
- (IV), (V)
, (V)* the equitable partitions of the -cube with quotient matrices *
\left(\begin{array}[]{ccc}0&c-1&n-c\\ c-1&0&n-c\\ c&c&n-2c-1\end{array}\right)* and \left(\begin{array}[]{ccc}c-1&n-c&0\\ c&n-2c-1&c\\ 0&n-c&c-1\end{array}\right), respectively*;**
- (VI)
the completely regular codes in with the intersection array .
By the definition, a completely regular code with the intersection array is a set of vertices such that the distance partition with respect to it is equitable with tridiagonal quotient matrix, and being the subdiagonal and the superdiagonal; so, the correspondence between (V) and (VI) is straightforward. The connection between (I) and (II) is noted in [19, 20]. (III), (IV), and (VI) are related in [16]. It is known [21, Proposition 2.3] that for odd , the arrays OA are in one-to-one correspondence with the self-complementary arrays OA (a set of vertices of is self-complementary if ); on the other hand, the array of type (II) must be self-complementary because of the distance invariance of equitable partitions, see e.g. [14].
Once, for and we have representatives of all equivalence classes of partitions of type (I), it is rather straightforward to find the number of equivalence classes of objects of types (II)–(VI).
Theorem 26**.**
There are exactly inequivalent orthogonal arrays OA. There are exactly inequivalent objects from each of the following families: orthogonal arrays OA; completely regular codes in with the intersection array ; equitable partitions of with quotient matrices and , respectively.
Remark 27*.*
Unifying the first two cells of a -partition with quotient matrix , , , we obtain an equitable -partition with quotient matrix . However, not all -partitions with quotient matrix can be obtained in such a way.
7 : discussion, connection with OA
The remaining quotient matrix related with equitable partitions of that attain the correlation-immunity bound and were not discussed in details above is . The corresponding equitable partitions are of some special combinatorial interest because the first cell of such partition induces a collection of disjoint cycles in the Hamming graph. We failed to make the complete classification of such partitions using any approach described here directly. However, the computational algorithm from Section 6 can be modified, dividing the classification into more steps depending on the value of some coordinate. This way can be successful, but requires relatively large amount of computational resources. Hopefully, the classification will be finished within several months, and at this moment we can only announce that there are more than equivalence classes of such partitions. In this section, we briefly discuss the length of cycles induced by such a partition and mention the relation with the quotient matrix , which corresponds to the orthogonal arrays OA.
7.1 Cycle lengths
As one can see from the first coefficient of the quotient matrix , the first cell of a corresponding equitable partition induces a regular subgraph of of degree , i.e., the union of disjoint cycles. Any theoretical information on the structure of a partition could simplify the classification; so, it is important to understand if the size of cycles is a constant or it can vary. In the following proposition, we show that there are partitions that induce both -cycles and -cycles. Another conclusion that can be made from it is that the simple construction [6, Prop. 1(c)] that multiplies the quotient matrix by an integer number can produce inequivalent equitable partitions if one varies the addition, treating the vertex set of the -cube as different modules.
Proposition 28**.**
For every from , there is an equitable partition of with quotient matrix such that the first cell induces cycles of length and cycles of length .
Proof.
(i) We start with and construct a required partition using the doubling construction [6, Proposition 1(c)]: if is an equitable partition of with quotient matrix , then is defined by
[TABLE]
Now consider an arbitrary vertex from . Let , and let be the only neighbor of from . Then , , , belong to and form a -cycle. So, every element of lies in a cycle of length with elements from .
(ii) Let . Again, we start with the partition and use the same construction but with different addition:
[TABLE]
Here, the sum of is defined by pairs of coordinates: , where : , , , is the Gray map from to GF. It is straightforward that the construction doubled the parameters of the quotients matrices for this modification as well as in the case of usual addition. Again, consider an arbitrary vertex such that and are from from . We see that , , , , , , , belong to and form a -cycle. So, every element of lies in a cycle of length with elements from .
(iii) For arbitrary , we use the same construction with the “mixed” addition, the addition in the first coordinates and the usual addition in the remaining ones. The first cell of consists of edges, edges of each direction. The edges of the first directions correspond to -cycles in , while the remaining edges correspond to -cycles. Counting the number of cycles is straightforward. ∎
7.2 and OA
The parameters of orthogonal arrays OA lie on the Bierbrauer–Friedman bound (2); hence, such arrays are simple. Moreover, as was observed in [19, 20], any array on this bound corresponds to an equitable -partition with the first coefficient of the quotient matrix being [math]. It is straightforward to see that the quotient matrix corresponding to OA is (indeed, and ); equitable partitions with this quotient matrix are known to exist [6, Proposition 2] and moreover, the recent classification result [16] says that they are all equivalent. By the argument similar to Remark 27, the “projection” of such a partition gives an equitable partition of with quotient matrix . It occurs that only (of more than ) inequivalent equitable partitions with quotient matrix are related with OA in such a way. Further studying of the exceptional properties of these three partitions can potentially give a tip how to construct other orthogonal arrays attaining bound (2) from equitable partitions. For example, putative OA are equivalent to putative equitable partitions with quotient matrix and related to equitable partitions with quotient matrix , which are known to exist (this matrix is a multiple of , considered in the current paper); so, one can try to construct OA starting from .
Appendix
Below we list all inequivalent equitable partitions with quotient matrix . The parameters are listed in the following order: rank, i.e., the dimension of the affine span of (, , or ); the order of the automorphism group, i.e., of the stabilizer of in ; the orbit sizes, for , then for ; the subspace (the “kernel”, given by a basis) and a set (the set of representatives of cosets of the kernel) such that (the kernel is the maximal subspace for which such decomposition is possible). The binary words of length are represented by hexadecimal numbers, e.g. .
Rank: 10, , orbits: ; ;
: 003, 005, 009, 030, 050, 090, 300, 500, 900, 111,
: {000}. 2. 2.
Rank: 11, , orbits: ; ;
: 300, 500, 900, 111, 222, 444, 888, 00f,
: {000, 003, 005, 081}. 3. 3.
Rank: 11, , orbits: ; ;
: 300, 500, 900, 111, 222, 444, 888, 003,
: {000, 005, 009, 048}. 4. 4.
Rank: 11, , orbits: ; ;
: 300, 500, 900, 111, 222, 444, 888,
: {000, 003, 005, 006, 00a, 00c, 00f, 018}. 5. 5.
Rank: 11, , orbits: ; ;
: 900, c00, 300, 444, 222, 099,
: {000, 003, 005, 006, 00a, 00c, 017, 018, 030, 03c, 04b, 050, 0a0, 0c0, 188, 809}. 6. 6.
Rank: 11, , orbits: ; ;
: 900, 300, 500, 144, 111, 0aa,
: {000, 003, 005, 006, 00a, 00c, 018, 01e, 027, 030, 060, 081, 096, 0c0, 488, 828}. 7. 7.
Rank: 11, , orbits: ; ;
: 900, c00, 300, 033, 066, 0cc,
: {000, 003, 006, 00c, 012, 018, 048, 069, 224, 428, 4e1, 805, 809, 80a, 811, 814}. 8. 8.
Rank: 11, , orbits: ; ;
: 900, 300, 500, 144, 4bb,
: {000, 003, 005, 006, 00a, 00c, 017, 018, 02e, 030, 035, 03c, 04b, 050, 059, 05a,
: {060, 069, 072, 081, 09c, 0a0, 0c0, 809, 811, 812, 821, 822, 828, 882, 888, 890}. 9. 9.
Rank: 11, , orbits: ; ;
: 900, 300, 500, 0aa, 055,
: {000, 003, 005, 006, 00a, 00c, 018, 027, 030, 036, 03c, 060, 06c, 081, 0b1, 0c0,
: {166, 2b4, 40f, 809, 811, 812, 814, 821, 822, 824, 828, 82d, 842, 848, 884, 890}. 10. 10.
Rank: 11, , orbits: ; ;
: 900, 300, 500, 847, 1b8,
: {000, 003, 005, 006, 00c, 018, 01b, 022, 02b, 02d, 030, 035, 048, 059, 05a, 060,
: {069, 071, 081, 08b, 090, 0c0, 809, 80a, 811, 812, 814, 821, 824, 850, 882, 884}. 11. 11.
Rank: 11, , orbits: ; ;
: 900, 300, 500, 0aa, 055,
: {000, 003, 006, 00c, 00f, 012, 018, 021, 030, 036, 039, 048, 060, 081, 084, 0c0,
: {1b1, 21e, 2cc, 472, 496, 805, 809, 80a, 811, 814, 822, 824, 828, 82d, 842, 890}. 12. 12.
Rank: 11, , orbits: ; ;
: 900, c00, 300, fff,
: {000, 003, 006, 00c, 00f, 011, 017, 018, 028, 02b, 02d, 030, 035, 036, 03a, 044,
: {04b, 04d, 053, 056, 059, 05a, 05c, 060, 063, 066, 06a, 071, 081, 082, 09a, 0c0,
: {155, 178, 1b1, 1c6, 247, 2cc, 41b, 46c, 472, 805, 809, 80a, 812, 814, 81d, 81e,
: {821, 822, 824, 82e, 841, 842, 848, 850, 884, 888, 88b, 890, 896, 8a0, 8c3, 8d8}. 13. 13.
Rank: 11, , orbits: ; ;
: 900, c00, 300, fff,
: {000, 003, 006, 00c, 00f, 011, 017, 018, 028, 02b, 02d, 030, 035, 036, 03a, 044,
: {04b, 04e, 053, 055, 059, 05a, 05c, 060, 063, 069, 072, 081, 082, 099, 09a, 0c0,
: {133, 1e4, 247, 256, 26a, 278, 427, 439, 46c, 4c3, 805, 809, 80a, 812, 814, 81d,
: {81e, 821, 822, 824, 82e, 841, 842, 848, 84d, 850, 871, 874, 884, 888, 890, 8a0}. 14. 14.
Rank: 11, , orbits: ; ;
: 900, c00, 300, fff,
: {000, 003, 006, 00c, 00f, 011, 018, 01d, 027, 028, 02d, 030, 033, 036, 03a, 044,
: {04b, 04e, 053, 055, 056, 059, 05a, 060, 063, 069, 06a, 081, 082, 08b, 09a, 0c0,
: {199, 21b, 21e, 22b, 235, 23c, 247, 278, 2a3, 46c, 472, 4b2, 805, 809, 80a, 812,
: {814, 817, 821, 822, 824, 82e, 841, 842, 848, 850, 871, 884, 888, 890, 8a0, 8c6}. 15. 15.
Rank: 12, , orbits: ; ;
: 111, 222, 444, 888, 003, 840,
: {000, 04c, 009, 005, 054, 090, 030, 060, 051, 066, 06a, 01d, 03a, 036, 02c, 07c}. 16. 16.
Rank: 12, , orbits: ; ;
: 00f, 0f0, f00, 333,
: {000, 005, 050, 500, 550, 505, 055, 555, 021, 028, 041, 048, 210, 280, 410, 480,
: {102, 802, 104, 804, 126, 146, 826, 846, 261, 461, 268, 468, 612, 614, 682, 684,
: {016, 086, 206, 406, 160, 860, 062, 064, 601, 608, 620, 640, 111, 118, 181, 811,
: {881, 818, 188, 888, 013, 083, 130, 830, 301, 308, 516, 586, 165, 865, 651, 658}.
Acknowledgements
The authors thank Evgeny Bespalov, Vladimir Potapov, and Olli Pottonen for useful discussions and the referees for helpful comments.
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