A Note on the LogRank Conjecture in Communication Complexity

Abstract

The LogRank conjecture of Lovász and Saks (1988) is the most famous open problem in communication complexity theory. The statement is as follows: suppose that two players intend to compute a Boolean function $f(x,y)$ when x is known for the first and y for the second player, and they may send and receive messages encoded with bits, then they can compute $f(x,y)$ with exchanging ${(\log \mathrm{rank}\left(M_f\right))}^c$ bits, where $M_f$ is a Boolean matrix, determined by function f. The problem is widely open and very popular, and it has resisted numerous attacks in the last 35 years. The best upper bound is still exponential in the bound of the conjecture. Unfortunately, we cannot prove the conjecture, but we present a communication protocol with ${(\log \mathrm{rank}\left(M_f\right))}^c$ bits, which computes a (somewhat) related quantity to $f(x,y)$. The relation is characterized by the representation of low-degree, multi-linear polynomials modulo composite numbers. Our result may help to settle this long-open conjecture.

1. Introduction

Two-player communication games were first defined by Yao in 1979 [1]. There are two players, Alice and Bob, and a Boolean function

$$f:{\{0,1\}}^N\times {\{0,1\}}^N\to \{0,1\},$$

and two Boolean sequences, $x,y\in {\{0,1\}}^N$. Alice knows the value of x, Bob the value of y, and they want to compute collaboratively the value of $f(x,y)$. The local computational power of the players is unlimited, and the cost of the collaborative computation is the number of bits exchanged between the parties. Function f is computed by the players if one of them knows $f(x,y)$, and the other knows that the first player knows [2].

Clearly, any f can be computed by N bits of communication; Alice tells x to Bob at cost N, and Bob computes $f(x,y)$ for free. This is the trivial protocol for computing f.

We say that the players follow a communication protocol for computing f, where the protocol prescribes, in each step, the message that a player should send to the other for a given message history and input. The cost of a protocol, computing $f(x,y)$, is the maximum number of bits communicated, taken for all x and y inputs.

The communication complexity of Boolean function f is the minimum cost of protocols, which compute f. The communication complexity of f is denoted by $\kappa \left(f\right)$ [2]. For a more formal introduction and examples, we refer to references [2,3].

The communication games and communication complexity have become a central field of theoretical computer science, hundreds of publications (e.g., references [4,5,6,7,8] and numerous books [3,9,10]) have appeared on the topic. For example, difficult-to-handle areas, such as Boolean circuit complexity, were applied to communication games for gaining the upper and lower bounds [10,11,12,13,14,15,16,17,18].

One of the main problems is finding the general upper and lower bounds for the communication complexity of f. To describe the bounds, we need to define the communication matrix of the function f:

Definition 1.

The communication matrix of $f:{\{0,1\}}^N\times {\{0,1\}}^N\to \{0,1\}$ is a $2^N\times 2^N$ 0-1 matrix $M_f$, where its rows correspond to the different $x\in {\{0,1\}}^N$ values, the columns to the different $y\in {\{0,1\}}^N$ values, and in the position of row x and column y is the value of $f(x,y)\in \{0,1\}$. Let $\mathrm{rank}\left(M_f\right)$ denote the matrix rank over the rational field. Let log denote the logarithm base 2, and ln the natural logarithm.

A general lower bound, which implies that, for most of the natural communication problems (e.g., the identity function or the set disjointness), the trivial protocol is optimal, which was proved by Mehlhorn and Schmidt in 1982:

Theorem 1

([19]). Suppose that f is not the identical 0 function. Then,

$$\kappa \left(f\right)\ge log\mathrm{rank}\left(M_f\right).$$

The most famous open problem in communication complexity is the LogRank conjecture of Lovász and Saks (1988):

Conjecture 1

([20]). There exists a polynomial P, such that for all $f\not\equiv 0$:

$$\kappa \left(f\right)\le P(log\mathrm{rank}\left(M_f\right)).$$

The 35-year-old conjecture is widely open today, inspiring numerous theorems and approaches even in the last few years [21,22,23,24,25,26,27].

The best published upper bound of Lovett [8] is still exponentially far from the conjectured upper bound, for large enough $\mathrm{rank}\left(M_f\right)$:

$$\kappa \left(f\right)=O\left(\sqrt{\mathrm{rank}\left(M_f\right)}log\mathrm{rank}\left(M_f\right)\right).$$

We note that the LogRank conjecture can be formulated with the terms of graph theory as a relation between the rank of the adjacency matrix and the chromatic number of a graph [8,20], without even mentioning communication games.

2. Preliminaries

We will need some definitions and theorems from references [28,29]:

Let $g:{\{0,1\}}^n\to \{0,1\}$ be a Boolean function and let m be a positive integer. Barrington, Beigel, and Rudich [30] gave the following definition:

Definition 2.

The polynomial P with integer coefficients weakly represents Boolean function g modulo m if there exists an $S\subset \{0,1,2,\dots ,m-1\}$, such that for all $x\in {\{0,1\}}^n$,

$$g\left(x\right)=0⟺\left(P\left(x\right)modm\right)\in S.$$

Here, $(amodm)$ denotes the smallest non-negative $b\equiv amodm$.

We are interested in the smallest degree polynomials representing g. Since g is Boolean, we may assume that P is multilinear (since $x_i^2=x_i$ over ${\{0,1\}}^n$).

Let ${\mathrm{OR}}_n:{\{0,1\}}^n\to \{0,1\}$ denote the n–variable OR-function:

$${\mathrm{OR}}_n(x_1,x_2,\dots ,x_n)=\left\{\begin{array}{c}0,\mathrm{if}{x}_1=x_2=\dots =x_n=0\hfill \\ 1\mathrm{otherwise}.\hfill \end{array}\right.$$

If polynomial P weakly represents ${\mathrm{OR}}_n$ modulo, a prime p, then we may assume that for $x\in {\{0,1\}}^n$,

$$P\left(x\right)=0modp\iff x=(0,0,\dots ,0).$$

Then,

$$1-P^{p-1}(1-x_1,1-x_2,\dots ,1-x_n)$$

is clearly the n-variable AND function with a unique multi-linear form

$$x_1{x}_2{x}_3\dots x_n.$$

Therefore, the degree of P is at least $n/(p-1)$.

Tardos and Barrington [31] showed that the same conclusion holds if p is a prime power.

However, Barrington, Beigel, and Rudich [30] proved that the conclusion fails for composite moduli with at least two prime divisors:

Theorem 2

([30]). There exists an explicitly constructible polynomial P of degree $O\left(n^{1/r}\right)$ which weakly represents $O{R}_n$ modulo $m=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_r^{{\alpha }_r}$, where the $p_i$’s are distinct primes.

An explicit example of such a low-degree, non-trivial polynomial modulo 6 is given in Section 4.

We have applied the polynomial of Theorem 2 in reference [32] for falsifying a long-standing conjecture for the size of set systems with restricted intersections and also for giving new explicit Ramsey graph constructions, among other applications for set systems and codes described in references [13,33,34,35,36,37].

In reference [32], we reproduced a short proof of Theorem 2 from reference [30], and we proved the following:

Corollary 1.

Let $m=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_r^{{\alpha }_r}$. Then, there exists an explicitly constructible polynomial $P^{\prime }$ with n variables and of degree $O\left(n^{1/r}\right)$ which is equal to 0 on $x=(0,0,\dots ,0)\in {\{0,1\}}^n$, it is nonzero mod m for all other $x\in {\{0,1\}}^n$, and for all $x\in {\{0,1\}}^n$ and for all $i\in \{1,\dots ,r\}$, $P\left(x\right)\equiv 0(mod{p}_i^{{\alpha }_i})$ or $P\left(x\right)\equiv 1(mod{p}_i^{{\alpha }_i})$.

Using the results of reference [32], we have found a remarkable application for elementary symmetric polynomials in reference [28].

We will need the following definition from reference [29] with small changes to describe the result:

Definition 3.

Let m be a composite number with prime factorization $m=p_1^{e_1}{p}_2^{e_2}\cdots p_{\ell }^{e_{\ell }}$. Let ${\mathbb{Z}}_m$ denote the ring of integers modulo m and $\mathbb{Z}$ the integers. Let f be a multi-linear polynomial of n variables over the integers:

$$f(x_1,x_2,\dots ,x_n)=\sum _{\alpha \in {\{0,1\}}^n}{a}_{\alpha }{x}^{\alpha },$$

where $a_{\alpha }\in \mathbb{Z}$, $x^{\alpha }={\prod }_{i=1}^n{x}_i^{{\alpha }_i}$. Then, we say that

$$g(x_1,x_2,\dots ,x_n)=\sum _{\alpha \in {\{0,1\}}^n}{b}_{\alpha }{x}^{\alpha },$$

is an

• alternative representation of f modulo m, if

  $$\forall \alpha \in {\{0,1\}}^n\exists j\in \{1,2,\dots ,\ell \}:$$

  $$a_{\alpha }\equiv b_{\alpha }(mod{p}_j^{e_j});$$
• 0-a-strong representation of f modulo m, if it is an alternative representation, and furthermore, if for some i, $a_{\alpha }\not\equiv b_{\alpha }(mod{p}_i^{e_i}),$ then $b_{\alpha }\equiv 0(mod{p}_i^{e_i});$
• 1-a-strong representation of f modulo m, if it is an alternative representation, and furthermore, if for some i, $a_{\alpha }\not\equiv b_{\alpha }(mod{p}_i^{e_i}),$ then $a_{\alpha }\equiv 0(modm).$

That is, for modulus 6 in the alternative representation, each coefficient is correct; either modulo 2 or modulo 3, but not necessarily both.

In the 0-a-strong representation, the 0 coefficients are always correct; both modulo 2 and 3. The non-zeros are allowed to be correct in either modulo 2 or 3, and if they are not correct, one of them, say 2, then they should be 0 mod 2. Consequently, the coefficient 1 can be represented by 1, 3, or 4, but nothing else.

In the 1-a-strong representation, the non-zero coefficients of f are correct for both moduli in g, but the zero coefficients of f can be non-zero either modulo 2 or modulo 3 in g, but not both.

Remark 1

([29]). The 1-a-strong representations of polynomial f can be written in the form modulo m:

$$f+p_1^{e_1}{g}_1+p_2^{e_2}{g}_2+\cdots +p_{\ell }^{e_{\ell }}{g}_{\ell },$$

where the $g_i$ have no monomials in common with each other, nor with f.

Example 1

([29]). Let $m=6$, and let $f(x_1,x_2,x_3)=x_1{x}_2+x_2{x}_3+x_1{x}_3$, then $g(x_1,x_2,x_3)=3x_1{x}_2+4x_2{x}_3+x_1{x}_3$ is a 0-a-strong representation of f modulo 6; $g(x_1,x_2,x_3)=x_1{x}_2+x_2{x}_3+x_1{x}_3+3x_1^2+4x_2$ is a 1-a-strong representation of f modulo 6; and $g(x_1,x_2,x_3)=3x_1{x}_2+4x_2{x}_3+x_1{x}_3+3x_1^2+4x_2$ is an alternative representation modulo 6.

Remark 2.

Clearly, the 1-a-strong representation of f is not unique. Suppose that

(i) 

all coefficients of f and $f^{\prime }$ are either 1 or −1 mod m, and

(ii) 

g is a 1-a-strong representation of f and also of $f^{\prime }$, where $f,f^{\prime }$ and g are multilinear, homogeneous degree-d polynomials, that is, every monomials of $f,f^{\prime }$ and g are degree d,

then $f=f^{\prime }$ modulo m. Clearly, one can set the monomials of f or $f^{\prime }$ to 1 one by one, and the $p_i$-multiplied monomials need to be 0 in g because of homogeneity.

We proved this in reference [28] and stated it in this form in reference [29] as follows:

Theorem 3

([28]). Let the prime factorization of a positive integer m be $m=p_1^{e_1}{p}_2^{e_2}\cdots p_{\ell }^{e_{\ell }}$, where $\ell >1$. Then, a degree-2 0-a-strong representation of the second elementary symmetric polynomial

$$S_n^2(x,y)=\sum _{\genfrac{}{}{0pt}{}{i,j\in \{1,2,\dots ,n\}}{i\ne j}}{x}_i{y}_j$$

(1)

modulo m:

$$\sum _{\genfrac{}{}{0pt}{}{i,j\in \{1,2,\dots ,n\}}{i\ne j}}{a}_{ij}{x}_i{y}_j$$

(2)

can be computed as the following product with coefficients from ${\mathbb{Z}}_m$:

$$\sum _{j=1}^{t^{\prime }}\left(\sum _{i=1}^n{b}_{ij}^{\prime }{x}_i\right)\left(\sum _{i=1}^n{c}_{ij}^{\prime }{y}_i\right)$$

where $t^{\prime }=exp\left(O\left(\sqrt[\ell ]{logn{(loglogn)}^{\ell -1}}\right)\right).$ Moreover, this representation satisfies that $\forall i\ne j:a_{ij}=a_{ji}$.

Now, we need the main theorem from reference [29]:

Theorem 4

([29], Theorem 6). Let $m=p_1^{e_1}{p}_2^{e_2}\cdots p_{\ell }^{e_{\ell }}$, where $\ell >1$, and $p_1,p_2,\dots ,p_{\ell }$ are primes. Then, a degree-2, 1-a-strong representation of the dot-product $f(x_1,x_2,\dots ,x_n,y_1,y_2,\dots ,y_n)={\sum }_{i=1}^n{x}_i{y}_i$ can be computed with $t=exp\left(O\left(\sqrt[\ell ]{logn{(loglogn)}^{\ell -1}}\right)\right)$ multiplications of the form

$$\sum _{j=1}^t\left(\sum _{i=1}^n{b}_{ij}{x}_i\right)\left(\sum _{i=1}^n{c}_{ij}{y}_i\right),$$

(3)

where all coefficients are integers.

The proof is immediate from Theorem 3 by subtracting the 0-a-strong representation of $S_n^2(x,y)$ from $(x_1+x_2+\dots +x_n)(y_1+y_2+\dots +y_n)$ [29], and $t=t^{\prime }+1$.

An explicit example for the $b_{ij}$ and $c_{ij}$ coefficients, with $m=6,n=16,t=13$, can be found in Section 4.

3. The LogRank Protocol

Theorem 5.

Let $n\ge 3$ be an integer, $m=p_1{p}_2\dots p_{\ell }$ be the product of the first $\ell =\lfloor loglogn\rfloor$ primes. Then, with ${(logn)}^{c_2}$ bits of communication, with a positive constant $c_2$, Alice and Bob can compute the value for any integer substitutions for $x=(x_1,x_2,\dots ,x_n)$ and any rational substitutions for $y=(y_1,y_2,\dots ,y_n)$ of the 1-a-strong representation g modulo m of the dot-product polynomial ${\sum }_{i=1}^n{x}_i{y}_i$:

$$g(x,y)=\sum _{i=1}^n{x}_i{y}_i+p_1{g}_1(x,y)+p_2{g}_2(x,y)+\dots +p_{\ell }{g}_{\ell }(x,y),$$

(4)

where for $i=1,2,\dots ,\ell$, polynomials $g_i(x,y)$ are homogeneous quadratic polynomials without common monomials.

Proof. 

Since $m=p_1{p}_2\dots p_{\ell }$ is the product of the first ℓ primes, $m=e^{(1+o(1\left)\right)\ell ln\ell }$, by the estimation of the first Chebyshev number [38]. Substituting $\ell =\lfloor loglogn\rfloor$ we ascertain that any non-negative integers up to m can be given with less than ${(loglogn)}^c$ bits, with a $c>0$.

For any integer substitutions for $x=(x_1,x_2,\dots ,x_n)$, Alice can compute privately and communicate for $j=1,2,\dots t$ the mod m values of sums from (3):

$$\left(\sum _{i=1}^n{b}_{ij}{x}_i\right)$$

(5)

for each j with ${(loglogn)}^c$ bits.

Since

$$t\le exp\left(O({(logn)}^{1/\ell }loglogn)\right)\le exp(c_{1}loglogn)\le {(logn)}^{c_1},$$

with a positive $c_1$, the total communication of Alice is polylogarithmic in n. Since Bob knows the values of $y_i$, for $i=1,2,\dots ,n$; he, privately, without any communication, computes the value of (3).

□

The following corollary is our main result here:

Corollary 2.

Suppose that we have a Boolean function $F:{\{0,1\}}^N\times {\{0,1\}}^N\to \{0,1\}$, with a $2^N\times 2^N$ 0-1 communication matrix $M_F$ with $\mathrm{rank}\left(M_F\right)=n$ over the rationals, $n\le 2^N$. By the definition of the rank, one may choose n linearly independent 0-1 columns of $M_F$, let the $2^N\times n$ 0-1 matrix X contain these columns. Then, there exists an $n\times 2^N$ rational matrix Y, such that $M_F=XY$. Now, any element of communication matrix $M_F$ is the dot product of a length-n row of X and a length-n column of Y. From Theorem 5, the substitution of the 1-a-strong representation of this dot product can be computed modulo m as with polylogarithmic communication in the rank n, i.e., with ${(logn)}^{c_2}$ bits of communication.

Here, we describe our protocol.

(i)

First, we may assume that both players know the rank decomposition of the communication matrix of F: $M_F=XY$.

(ii)

Alice substitutes the integer values of the $x_i$ numbers for $i=1,2,\dots ,n$ into (5) for each j, and communicates the mod m values of the (5) sums for $j=1,2,\dots ,t$ to Bob.

(iii)

Bob knows the values of $y_i$, for $i=1,2,\dots ,n$. Consequently, he, privately, without any communication, computes the value of (3).

Communication occurs only in step (ii), when $t\le {(logn)}^{c_1}$ mod m values are transmitted, each with ${(loglogn)}^c$ bits.

Remark 3.

For a given $M_f$, the players can agree that they will compute the $kF(x,y)$ instead of $F(x,y)$, and then matrix Y can be changed for an integer matrix $kY$. Note also that this convention will not increase the communication, since Bob does not communicate the linear combinations of his variables.

4. Remarks and Examples

Note that the “surplus” terms in (4) are zero modulo for at least one of the prime divisors of m.

Remark 4.

We do not know how to eliminate the surplus terms with the $g_i$ polynomials from $\left(4\right)$ with a log-rank bounded additional communication. One can imagine numerous possible approaches, for example, substituting a prime $p_i,1\le i\le \ell$ for all or just a subset of 1s in the $x_i$s, and repeating the protocol, repeating the protocol above for 0-1 rows instead of 0-1 columns, or repeating the protocol above for overlaying subsets of indices $i=1,2,\dots ,n$ several times.

Remark 5.

As we described in reference [29], Theorem 4 has numerous applications in representing the matrix product with very few multiplications or the hyperdense coding of numbers, vectors or matrices. We also note that our definition and use of the term "hyperdense coding" [39] precedes the quantum-computational use of an unrelated but identically named term of [40,41] by more than 9 years.

Here, we give an example for a polynomial that we presented in reference [32]:

Example 2

([32]). Let $m=6$, and let

$$G_1\left(x\right)=\sum _{j=1}^{2^3-1}{(-1)}^{j+1}{s}_j\left(x\right),$$

and

$$G_2\left(x\right)=\sum _{j=1}^{3^2-1}{(-1)}^{j+1}{s}_j\left(x\right).$$

Then,

$$P\left(x\right)=3G_1\left(x\right)+4G_2\left(x\right)$$

weakly represents $O{R}_{71}$ modulo 6, and its degree is only 8.

Example 3.

Here, we give an explicit example for the coefficients in Theorem 4 (3).

Let $m=6,n=16,t=13$ and let B be a $16\times 13$ matrix, C be a $13\times 16$ matrix mod 6, then $A=BC$ mod 6 is a $16\times 16$ matrix with 1’s in the main diagonal and 0s either mod 2 or mod 3 or both outside the main diagonal:

$$B=\left[\begin{array}{ccccccccccccc}1& 0& 1& 1& 4& 1& 4& 4& 3& 1& 4& 4& 4\\ 1& 1& 0& 4& 1& 4& 1& 3& 4& 4& 1& 3& 3\\ 1& 1& 4& 0& 1& 4& 3& 1& 4& 4& 3& 1& 3\\ 1& 4& 1& 1& 0& 3& 4& 4& 1& 3& 4& 4& 4\\ 1& 1& 4& 4& 3& 0& 1& 1& 4& 4& 3& 3& 1\\ 1& 4& 1& 3& 4& 1& 0& 4& 1& 3& 4& 4& 4\\ 1& 4& 3& 1& 4& 1& 4& 0& 1& 3& 4& 4& 4\\ 1& 3& 4& 4& 1& 4& 1& 1& 0& 4& 3& 3& 3\\ 1& 1& 4& 4& 3& 4& 3& 3& 4& 0& 1& 1& 1\\ 1& 4& 1& 3& 4& 3& 4& 4& 3& 1& 0& 4& 4\\ 1& 4& 3& 1& 4& 3& 4& 4& 3& 1& 4& 0& 4\\ 1& 3& 4& 4& 1& 4& 3& 3& 4& 4& 1& 1& 3\\ 1& 4& 3& 3& 4& 1& 4& 4& 3& 1& 4& 4& 0\\ 1& 3& 4& 4& 3& 4& 1& 3& 4& 4& 1& 3& 1\\ 1& 3& 4& 4& 3& 4& 3& 1& 4& 4& 3& 1& 1\\ 1& 4& 3& 3& 4& 3& 4& 4& 1& 3& 4& 4& 4\end{array}\right]$$

$$C=\left[\begin{array}{cccccccccccccccc}1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 1& 0& 0& 1& 1\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 1& 0& 1& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& -1& 0& -1& -1& -2\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0& 1& 0& 1& 1& 2\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& -1& 0& -1& 0& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& -1& 0& 0& -1& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& -1& -1& -1\end{array}\right]$$

$$A=BC=\left[\begin{array}{cccccccccccccccc}1& 0& 0& -3& 0& -3& -3& -2& 0& -3& -3& -2& -3& -2& -2& -3\\ 0& 1& -3& 0& -3& 0& -2& -3& -3& 0& -2& -3& -2& -3& -3& -2\\ 0& -3& 1& 0& -3& -2& 0& -3& -3& -2& 0& -3& -2& -3& -3& -2\\ -3& 0& 0& 1& -2& -3& -3& 0& -2& -3& -3& 0& -3& -2& -2& -3\\ 0& -3& -3& -2& 1& 0& 0& -3& -3& -2& -2& -3& 0& -3& -3& -2\\ -3& 0& -2& -3& 0& 1& -3& 0& -2& -3& -3& -2& -3& 0& -2& -3\\ -3& -2& 0& -3& 0& -3& 1& 0& -2& -3& -3& -2& -3& -2& 0& -3\\ -2& -3& -3& 0& -3& 0& 0& 1& -3& -2& -2& -3& -2& -3& -3& 0\\ 0& -3& -3& -2& -3& -2& -2& -3& 1& 0& 0& -3& 0& -3& -3& -2\\ -3& 0& -2& -3& -2& -3& -3& -2& 0& 1& -3& 0& -3& 0& -2& -3\\ -3& -2& 0& -3& -2& -3& -3& -2& 0& -3& 1& 0& -3& -2& 0& -3\\ -2& -3& -3& 0& -3& -2& -2& -3& -3& 0& 0& 1& -2& -3& -3& 0\\ -3& -2& -2& -3& 0& -3& -3& -2& 0& -3& -3& -2& 1& 0& 0& -3\\ -2& -3& -3& -2& -3& 0& -2& -3& -3& 0& -2& -3& 0& 1& -3& 0\\ -2& -3& -3& -2& -3& -2& 0& -3& -3& -2& 0& -3& 0& -3& 1& 0\\ -3& -2& -2& -3& -2& -3& -3& 0& -2& -3& -3& 0& -3& 0& 0& 1\end{array}\right]$$

If we take the dot product of column j of B each with $(x_1,x_2,\dots ,x_n)$, then we will obtain the t left sums of (3):

$$\left(\sum _{i=1}^n{b}_{ij}{x}_i\right)$$

If we take the dot product of row j of C each with $(x_1,x_2,\dots ,x_n)$, then we will obtain the t right sums of (3):

$$\left(\sum _{i=1}^n{c}_{ij}{y}_i\right)$$

Now, clearly, in (3) one can obtain the coefficient of $x_u{y}_v$ by computing the dot product of row u of B with column v of C; that is, matrix A describes the coefficients of the mod 6 representation of the dot product $f(x_1,x_2,\dots ,x_n,y_1,y_2,\dots ,y_n)={\sum }_{i=1}^n{x}_i{y}_i$; all the elements of the main diagonal are 1, and the others are 0 mod 2 or mod 3 or both.

For readers wishing to experiment with these matrices, a Maple worksheet can be accessed at https://grolmusz.pitgroup.org/?attachment_id=1840, accessed on 3 October 2023.