1 Introduction

The concept of F-squares is a generalisation of Latin squares, first introduced by MacMahon [1] in 1898 under the name “quasi-Latin squares.” Subsequently, F-squares have been examined both directly and indirectly by Finney [2–4], Freeman [5], and Addelman [6] as a secondary outcome of their research interests in experimental designs. The first formal definition and use of the term “F-square” is due to Hedayat, who also established foundational results on their properties, motivated by practical and theoretical needs in the theory of designs [7,8].

Let be positive integers and . A frequency square or more concisely an F-square, of type and order n is an array over such that the each symbol appears exactly times in each row and in each column. In the case where , the square is said to be type [9]. An F-square of type (n; 1) corresponds to a Latin square of order n. When the symbols represent treatments, a square of type serves as an experimental design with the properties: (a) treatment effects are orthogonal to row effects; (b) treatment effects are orthogonal to column effects; and, if , then (c) treatment arrangement is balanced within rows and columns; and (d) row effects are orthogonal to column effects [8]. There are two main reasons for considering F-square designs over Latin square designs in experimentation. First, when the difference between some treatments considered in previous experimentation as distinct is negligible and could be ignored. Second, when the number of treatments is smaller than the order of the square.

Recent research highlights a wide range of applications of F-squares in experimental design. Aggarwal et al. have explored mixture designs in orthogonal blocks using F-squares [10], nearly optimal orthogonally blocked designs for four mixture components based on F-squares [11], orthogonally blocked mixture component–amount designs via projections of F-squares [12], and optimal orthogonal block designs for four-mixture components in two blocks, based on F-squares for Becker’s models and K-models [13]. Husain et al. have further investigated optimal orthogonal designs in two blocks based on F-squares for the mixture inverse model with four components [14], F-square-based four-component D-, A-, and E-optimal orthogonal designs for an additive quadratic mixture model [15], uniform designs based on F-squares [16], and F-square-based optimal designs for reduced cubic canonical models with four components [17].

Research in the theory of F-squares primarily centres on constructing sets of mutually orthogonal F-squares (MOFS). Two F-squares, F₁ and F₂, of type are said to be orthogonal if each ordered pair (i, j) occurs times when the squares are superimposed. A MOFS is a set of F-squares in which each pair is orthogonal. A set of MOFS of cardinality k, each of type , is denoted by k-. Hedeyat et al. [18] showed that a set of type k-MOFS contains at most squares. Such a set is called complete. The only complete set of MOFS known are of type with either: (i) n is a prime power; or (ii) if there exits a Hadamard matrix of order n for [9]. Mavron [19] described complete sets of MOFS based on a technique for constructing affine 2-designs with parallel classes of lines. A set of k-MOFS is said to be maximal, denoted by k-, if there does not exist a F-square of type that is orthogonal to every for all and . For even n, let be the smallest k such that there exists a set of k-. It was shown in Britz et al. [20] that if is odd and if is even. This result was extended by Cavenagh et al. [21] who showed if is even, then .

This paper investigates the conditions under which types of F-squares can be generated by cellular automata (CAs). A cellular automaton (CA) is a discrete dynamical system described by a regular lattice of cells, where each cell synchronously updates its value by applying a local rule to itself and its neighbouring cells. There has been considerable attention into the relationship between CAs and Latin squares, especially concerning CAs defined over finite fields [22–25]. In particular, the survey by Manzoni, Mariot and Menara [26] offers a comprehensive account of how combinatorial designs can be constructed via CAs, with a particular focus on mutually orthogonal Latin squares, and also reviews the applications of CA-based designs in areas such as cryptography and coding theory. Our study is motivated by the work of Mariot et al. [25], who found a construction of Latin squares generated by CAs.

The paper is organised as follows. Section 2 covers the necessary background on F-squares, cellular automata and prior research. Section 3 presents our main results. Finally, Section 4 summarises the key contributions of this study and explores promising directions for future research.

2 Background

In this section, we summarise some definitions, lemmas, and theorems that are used in our results. For further reading, we refer the reader to [9] and [25].

3 Main results

Throughout, we assume that we have blocked together sites of the underlying CA, , with local rule is f of diameter d to produce a d = 2 blocked CA, whose local rule is a binary operation. The elements of are always assumed to be lexicographically ordered unless stated otherwise. The elements of the square generated by the CA are illustrated in Table 1.

The sum of any row i (denoted by ) and the sum of any column j (denoted by ) in the square generated by the CA are given as follows:

Since the elements of are arranged in lexicographically order, the th and the (2m)th elements differ only in their final bit while the (m)th and the th elements differ only in their first bit for .

Now we consider the condition under which the column sums of columns and (2m), as well as the row sums of rows (m) and , are equal. This restrictive case is important because, in an F-square from a CA, every column sums and row sums are equal; therefore, the column/row sum of any pair of columns/rows must also be equal.

Define, for , the operation to denote integer (not modulo 2) addition. We extend this notation to sums of multiple -valued terms, where all additions are interpreted over .

Lemma 2. Let be a CA defined by a local rule . Then the column sums of the th and (2m)th columns in the square generated by a CA are equal if and only if

where and .

Proof. Consider two columns, th and (2m)th indexed by and . Without loss of generality we may take and , which differ only in their last bit. Let and C_2m be the respective column sums:

Similarly,

Note that all components of and C_2m are identical except possibly the last, which depends on whether the final bit is 0 or 1. Now observe that for fixed , the final argument of f varies only in the first component , which ranges over all binary vectors. In particular, , and both values appear exactly times among the . Therefore:

From :

From C_2m:

Now, we conclude that C_2m+1 and C_2m+2 are equal if and only if:

(1)

We now examine the corresponding condition for row sums in the square generated by a CA. □

Lemma 3. Let be a CA defined by a local rule . Then the row sums of the (m)-th and -th rows in the square generated by CA are equal if and only if

where and .

Proof. Consider two rows, (m)th and th indexed by and . Without loss of generality we may take and , which differ only in their first bit. Let and be the respective row sums:

Similarly,

Note that all components of and are identical except possibly the first, which depends on whether the first bit is 0 or 1. Now observe that for fixed , the final argument of f varies only in the last component x_1,j, which ranges over all binary vectors. In particular, , and both values appear exactly times among the . Therefore:

From :

From :

Now, we conclude that and are equal if and only if:

(2)

□

From this point forward, we will refer to pair of columns that differ only in their final (rightmost) bit, for example, the st and (2m)nd columns, as paired columns. Similarly, we refer to pair of rows that differ only in their first (leftmost) bit, such as the (m)th and th rows, as paired rows.

We have established two conditions under which paired columns sums and paired rows sums in a square generated by a CA are equal. We now turn our attention to identifying the local rules of CAs over that satisfy those conditions.

Lemma 4. Let be a CA defined by a local rule . Then, the column sums of paired columns in the square generated by CA are equal if and only if the local rule f is of the form:

or

for some functions .

Proof. Any Boolean function can be expressed by fixing a variable and decomposing the function in terms of the remaining variables. In particular, by fixing , the local rule f can be written as:

(3)

for some functions .

We can further decompose each for i = 1,2 by fixing x₁ and expressing it in terms of functions of variables. This yields:

(4)

for some functions . To determine the column sum of paired columns are equal, we consider the condition in Eq (1) of Lemma 2, which involves evaluating the local rule at four points. Take any . Using the decomposition in Eq (4), we evaluate:

(5)(6)(7)(8)

From Eq (1) and the evaluations in (5)–(8), we derive:

(9)(10)

All functions and are evaluated at . Note that each term in Eq (1) is an element of , i.e., either 0 or 1. For the function f to yield equal column sums, Eq (9) and Eq (10) must evaluate to the same value. We proceed by analysing two cases based on this value according to the Table 2.

If , where , then the pair and must contain the same multiset of values. This occurs in exactly two cases:

• Case I: and
• Case II: and

Note that the two cases overlap precisely when . That is, both cases apply when all four variables are equal – either all 0 or all 1.

1. Case I:

We are given the following congruences modulo 2:

(11)(12)

From Eq (11):

Substituting g₂ = 0 into Eq (12), we obtain

Therefore, both conditions are satisfied if and only if:

Now, since range over all values in , the equations above hold for all such values. Hence, substituting these into Eq (4), we obtain:

1. Case II:

(13)(14)

From Eq (14):

(15)

From Eq (13) and Eq (15):

Now, since range over all values in , the equations above hold for all such values. Hence, substituting these into Eq (4), we get:

□

Lemma 5. Let be a CA defined by a local rule . Then, the row sums of paired rows in the square generated by CA are equal if and only if the local rule f is of the form:

or

for some functions .

Proof. Similar to the proof of Lemma 4, to determine when paired row sums are equal, we consider the row sum condition (see Eq (2)), which involves evaluating the local rule at four points. Take any . Using the decomposition in Eq (4), we evaluate:

(16)(17)(18)(19)

Similarly, from Eq (2) and the evaluations in (16)–(19), we derive:

(20)(21)

All functions and are evaluated at . Note that each term in Eq (2) is an element of , i.e., either 0 or 1. For the function f to yield equal row sums, Eq (20) and Eq (21) must evaluate to the same value. We proceed by analysing two cases based on this value.

If , where , then the pair and must contain the same multiset of values. This occurs in exactly two cases:

• Case I: and
• Case II: and

Note that the two cases overlap precisely when . That is, both cases apply when all four variables are equal – either all 0 or all 1.

1. Case I:

We are given the following congruences modulo 2:

(22)(23)

From Eq (22):

Substituting h₁ = 0 into Eq (23), we obtain

Therefore, both conditions are satisfied if and only if:

Now, since range over all values in , and the equations above hold for all such values. Hence, substituting these into Eq (4), we obtain:

1. Case II:

We are now given the following congruences modulo 2:

(24)(25)

From Eq (25):

(26)

From Eq (24) and Eq (26):

Now, since range over all values in , and the equations above hold for all such values. Hence, substituting these into Eq (4), we get:

□

We now examine the identification of local rules that simultaneously yield paired rows with equal row sums and paired columns with equal column sums.

Lemma 6. Let be a CA defined by a local rule . Then, the column sums of paired columns and the row sums of paired rows in the square generated by CA are equal simultaneously if and only if the local rule f is of one of the following forms:

or

where .

Proof. From Lemma 4, the column sums of paired columns in the square generated by are equal if and only if the local rule f is of the form

From Lemma 5, the row sums of paired rows are equal if and only if

The common form of the local rule for both Lemma 4 and Lemma 5 is:

which simultaneously satisfies both conditions and represents one possible form of the local rule functions.

Now, consider the case where . This corresponds to choosing in Lemma 4 and in Lemma 5, so that

Thus, the function satisfies the structural conditions in both lemmas and represents the second possible form of the local rule functions. □

Lemma 6 establishes that for squares generated by CAs, the column sums of any paired columns and the row sums of any paired rows are equal if the local rule f takes one of the following forms:

(27)

For convenience, let us denote the two local rule functions as Type 1 and Type 2, respectively.

Since a necessary condition for a square to be a F-square is that all row and column sums are equal, it is natural to ask whether such local rules always generate F-squares. To investigate this, we present two counterexamples for each type when d = 3, demonstrating that the local rules do not, in general, produce F-squares.

Two squares and , shown in Figs 3 and 4, respectively, are not F-squares. More importantly, we observe that in , the paired columns and paired rows are identical. This occurs because the first case, g₁, is independent of the leftmost and rightmost variables of f. However, the sums of non-paired columns and rows differ, indicating that uniformity in sums is not guaranteed beyond paired positions. This suggests a need to investigate further which form of Type 1 local rules produce squares with equal column and equal row sum.

Proposition 1. Let be a CA defined by the local rule given by

Then the column sums of every pair of two columns and the row sums of every pair of two rows in the square generated by CA are equal if and only if

for some constant .

Proof. First, we consider the column sum requirement. Let two distinct columns be indexed by and , where . The column sums are then be given by:

Without loss of generality, we may assume that all components of and are identical except possibly the last, namely and :

This calculation yields equal only if for all inputs, which implies that g₁ is constant:

In the similar way, we can work on the row sum requirement as well. Thus, we conclude that

Moreover, if is constant, then the output of the CA is uniform, and every entry in the resulting square is the same. Therefore, every row and every column has the same sum. This completes the proof. □

In contrast, in , although column sums of any paired columns and the row sums of any paired rows are equal, the entries themselves differ. For instance, the entries in the third column are {2,1,2,3}, while those in the forth column are {1,2,1,4}.

In addition to having equal sums for paired columns and paired rows, it is also necessary that the paired columns and paired rows contain the same elements. The local rules of Type 2 arise from the second cases of Lemma 4 and Lemma 5. Therefore, an additional condition must be imposed in the second case to ensure that paired rows and paired columns have the same multiset of elements.

Proposition 2. Let be a CA defined by the local rule given by

Then, the column sums of paired columns containing the same multiset of elements and the row sums of paired rows containing the same multiset of elements in the square generated by CA are equal simultaneously if and only if

Proof. First consider Case II of Lemma 4, where the condition is and . Here we have the additional requirement that or . This is necessary because, as illustrated in Fig 4, we observe that in some paired columns, , while the other paired columns, and with , which disturbs consistency across the columns. Therefore, we require or to maintain consistency. Note that we do not consider the case here, as it overlaps with Type 1. Without loss of generality, we may assume . Thus, we have:

(28)(29)(30)

From Eq (29):

(31)

From Eq (28) and Eq (31):

From Eq (30):

Now, since range over all values in , the equations above hold for all such values. Hence, substituting these into Eq (4), we get:

We can achieve the same result by considering Case II of Lemma 5.□

Now, let us consider the local rule functions identified previously. From Proposition 1, we have the local rule function , where . From from Proposition 2, the local rule function is , which represents bipermutive local rules. Thus, we have found that equal row and column sums are achieved when the local rule is either a constant function or bipermutive. When the local rule is , all entries of the square generated by CA are the same, resulting in a trivial F-square. When the local rule is bipermutive, the square is a Latin square. Therefore, we arrive at the following main result:

Theorem 1. Let be a CA defined by a local rule . Then, the F-square generated by CA is either:

• a trivial square (type ), or
• a Latin square (type ).

This result highlights a specific limitation of the binary setting. The entries of the CA-generated square are symbols drawn from an alphabet of size , obtained by encoding vectors over , so that each coordinate of a symbol takes the value 0 or 1. The F-square condition requires that each symbol occurs with the same total frequency in every row and every column. Over , the equal‑frequency requirement, together with the constraints imposed by the underlying local rule, leads to a highly restricted class of behaviours: either the symbols are distributed uniformly across all rows and columns, so that the resulting array is a Latin square, or the square is trivial, with each row and column taking a constant value. In other words, there are no “intermediate” frequency patterns arising from binary CAs in this setting.

4 Conclusion

In this paper, we investigated the existence of F-squares generated by CAs over . Our primary approach was to analyse the row sums and column sums of the square produced by the CA, noting that these sums are equal in a F-square. We directed our attention to specific pairs of columns and rows, namely, pair of columns whose binary representation differ only in their final (rightmost) bit (for example, the th and (2m)th columns), refereed to as paired columns, and pair of rows whose binary representation differ only in their first (leftmost) bit (such as the (m)th and th rows), refereed to as paired rows. These pairs were selected because they provide a focused and tractable basis for analysis. Moreover, ensuring that all such pairs have equal sums is a necessary condition for constructing an F-square from a CA.

We established two conditions under which paired columns sums and paired rows sums in a square generated by a CA are equal. We then focused on identifying the local rules of CAs over that satisfy these conditions. In particular, we examined the local rules that simultaneously yield paired columns with equal column sums and paired rows with equal row sums, and identified two types: Type 1, defined by , and Type 2, defined by . For each type, we presented counterexamples (when d = 3) demonstrating that, in general, these local rules do not always produce F-squares. This highlighted the necessity to further investigate which forms of Type 1 and Type 2 local rules do guarantee equal row and column sums. Our findings revealed that, for Type 1, when , where , and for Type 2, when , the required condition achieved. These specific families of local rules correspond to the constant local rules and bipermutive local rules, respectively, leading to the conclusion that the F-squares generated from CAs over are either trivial F-squares or Latin squares.

From a practical point of view, this result is still informative for applications in experimental design. The works of Aggarwal et al. [10–13] and Husain et al. [14–17] show that F-squares are used to construct a variety of mixture designs. Our analysis indicates that, when we restrict attention to CAs over , such designs can only be obtained from trivial F-squares or Latin squares. Since Latin squares and trivial F-squares are themselves valid frequency squares, this means that binary CA constructions can be employed as an alternative way to generate designs of the types considered in these studies, particularly when the number of treatments matches the order of the square. On the other hand, our results indicate that binary CA constructions naturally align with Latin-square-based structures already used in cryptography, such as those appearing in threshold secret sharing schemes and in the use of CAs to construct bent and correlation-immune Boolean functions and pseudorandom sequences [26].

An immediate avenue for future research is to explore the necessary conditions under which F-squares can be generated by CAs defined over arbitrary finite fields , where q > 2. For such CAs, the entries of the CA-generated square are symbols drawn from an alphabet of size , obtained by encoding vectors over . In this case, each coordinate of a symbol can take q values which is greater than 2 (the binary case), and there are many more ways to distribute these symbols while still keeping their total frequencies equal across all rows and columns. This suggests that non‑trivial, non‑Latin F-squares could in principle arise from suitable CAs over . Based on the case of q = 2, we can conjecture that there exist at least F-squares of types and . A natural next step is to examine whether the observed pattern persists for q = 3.

For , we consider the simplest diameter, namely d = 2. Let be a CA defined by the local rule given by

The associated square, shown in Fig 5, is an F-square of type (3;2,1), and it is neither a Latin square nor a trivial F-square.

This example illustrates that, for q = 3, the local rule acts on a larger alphabet, and the equal-frequency condition on rows and columns can be satisfied by symbol distributions that are neither fully Latin nor trivial. It is therefore plausible that further non-trivial, non-Latin F-squares might exist for CAs over . This investigation could benefit from a formal definition of potential non-trivial, non-Latin, F-squares. The ultimate goal would be to determine whether such F-squares exist and, if so, to characterise their structure and corresponding local rules. This generalisation introduces new structural and algebraic challenges, particularly in understanding how the local rule behaves over a larger alphabet and how it influences the combinatorial properties of the resulting squares.

Acknowledgments

The authors would like to thank the academic editor and the two anonymous reviewers for their useful comments and suggestions that highly improved the quality of this paper.