As the oldest known combinatorial designs, magic squares and their various variants have been applied to many fields, such as engineering technology [1,2,3,4,5], cryptography [6], information hiding [7,8], physics [9], pedagogy [10], number theory [11], matrix algebra [12]. Magic squares and their many relatives can be viewed as matrix representations of magic labelings of related set systems or hypergraphs [13,14] and were used to construct some labelings of graphs [15,16,17,18,19].

Given two positive integers n and d and let $I_n=\{0,1,\cdots ,n-1\}$. Let A be a d-dimensional matrix of side or order n consisting of integers with entries $a\left(x\right)$ or $a_{i_1,\cdots ,i_d}$, where $x\in I_n^d$ or $i_1,\cdots ,i_d\in I_n$. A row or line or hyperedge of A is an n-tuple of positions or points $(i_1,\cdots ,i_d)$ such that the number of identical coordinates for any two points is exactly $d-1$. A (space) diagonal is an n-tuple of points $\{(x,i_2\cdots ,i_d):i_k=x\mathrm{or}n-1-x,2\le k\le d,x\in I_n\}$. Let ${\mathbb{L}}_n^d$ denote the set of $d{n}^{d-1}$ rows and $2^{d-1}$ diagonals of any d-dimensional matrix of side n. Obviously, $(I_n^d,{\mathbb{L}}_n^d)$ is an n-uniform hypergraph or a set system. Similar to the definition for a magic labeling of a set system [13], we give the following definitions. A magic labeling of the hypergraph $(I_n^d,{\mathbb{L}}_n^d)$ is a labeling a of its points by integers such that every line has the same sum, that is, ${\sum }_{x\in L}a\left(x\right)=c$ for some constant c and $L\in {\mathbb{L}}_n^d$; we call the constant c the magic sum. Let $A=\left(a\right(x\left)\right)$ for $x\in I_n^d$, then the matrix A can be thought of as the label matrix of the above labeling a of the hypergraph $(I_n^d,{\mathbb{L}}_n^d)$. We call the matrix A a d-dimensional magic hypercube of side or order n. Other types of magic hypercubes can be defined similarly. A d-dimensional magic hypercube with side n is normal if it consists of $n^d$ consecutive integers. Usually, a 2-dimensional magic hypercube of order n is called a magic square of order n (MS$\left(n\right)$). The existence problem for normal magic hypercubes has been completely solved [20,21].

In this paper, we are interested in investigating multimagic squares. Let n, d, and t be positive integers. Let A and B be integer matrices with the same size and let $A*B$ denote their Hadamard product, with $(i,j)$ entry $a_{i,j}{b}_{i,j}$. Based on the reference [22], we write $A^{*d}=\left(a_{i,j}^d\right)$; obviously, $A^{*d}$ is the Hadamard product $A*\cdots *A$ (where the matrix A appears d times). Now let A has order n, then A is a t-multimagic square (MS$(n,t)$) if for $d\in \{1,\cdots ,t\}$, $A^{*d}$ is a magic square. We denote by an NMS$(n,t)$ a normal MS$(n,t)$. Usually, we call an MS$(n,2)$ a bimagic square and an MS$(n,3)$ a trimagic square. Many researchers have done a lot of work on the existence and construction of normal multimagic squares [23,24,25,26,27,28,29,30].

We now review the research on normal trimagic squares with even orders. In 2007, Derksen et al. [22] proved that there is no an NMS$(2n,3)$ for every positive odd integer n. In 2023, Hu et al. [25] proved that an NMS$(16n,3)$ exists for all positive integers n. However, very little is known for the existence of trimagic squares of doubly even orders not divisible by 16 except for the following three examples and three families. In 2017, based on known bimagic squares and trimagic squares of orders 12, 24, and 40, Li et al. [31] showed that there is an NMS$(mn,3)$ for $m\in \{12,24,40\}$ and infinitely many odd positive integers n. Therefore, the existence problem for normal trimagic squares of even orders is at present far from being solved.

This paper aims to construct new infinite families for normal trimagic squares of even orders. Our construction tools include a known product construction (see Section 2, Lemma 2), two new product constructions (see Section 3, Theorem 1 and Lemma 9), trimagic pairs, and (quasi-normal) row-square magic rectangles (new notions, see Section 2 for the definitions).

The outline of the rest of this paper is as follows. Section 2 presents some related preliminaries, including notations, notions, and lemmas. In Section 3, we firstly show that one can construct an NMS$(n,3)$ based on a trimagic pair of orthogonal diagonal Latin squares as well as an NMS$(pqr,3)$ based on normal $p\times q$ and $r\times q$ row-square magic rectangles for appropriate positive integers $p,$q, and r, then we prove several lemmas and theorems for the existence of quasi-normal even row-square magic rectangles and construct their several new infinite classes, finally, we provide new families of trimagic squares of even orders. In Section 4, we give conclusions of the work.

Let T be an n-set. For any $m\times n$ integer matrix A, let $\left|A\right|=\left(\right|a_{i,j}\left|\right)$, where $|a_{i,j}|$ denotes the absolute value of $a_{i,j}$; we denote by ${\mathfrak{S}}_A$ the entry-set of A and index the rows by $I_m$ and columns by $I_n$. An $m\times m$ array A is a diagonal Latin square over the set T with order m (DLS$\left(m\right)$) if

for $k\in I_m$. Two DLS$\left(m\right)$s C and D are called orthogonal if $\{(c_{i,j},d_{i,j}):i,j\in I_m\}={\mathfrak{S}}_C\times {\mathfrak{S}}_D$. The following can be found in [32].

Let C and D be two orthogonal DLS$\left(n\right)$s over $I_n$. The pair $(C,D)$ is a trimagic pair (TMP$\left(n\right)$) if $C*D$, $C^{*2}*D$, and $C*D^{*2}$ are all magic squares.

A row-magic rectangle (RMR) is an integer matrix A having the property that the row sums are all a constant. If the transpose of A is also a row-magic rectangle, then we call A a magic rectangle (MR). If $A^{*2}$ is also a row-magic rectangle, then we call A a bimagic row-magic rectangle (BRMR$(m,n)$). We call a BRMR$(m,n)$A a row-square magic rectangle (RSMR$(m,n)$) if A is a magic rectangle. Similarly, one can define a t-multimagic rectangle. A magic rectangle is normal if it consists of consecutive integers. Similar to a bimagic square, we call a 2-multimagic rectangle a bimagic rectangle. A bimagic rectangle and its transpose are row-square magic rectangles, but not converse. For results of multimagic rectangles, the interested reader is referred to the references [33,34].

Let $J_{m\times n}$ be an $m\times n$ all ones matrix. For integers c and d with $c\le d$ and $c\equiv d(mod2)$, let ${[c,d]}_2$ denote the set $\{c,c+2,\cdots ,d-2,d\}$. A BRMR$(p,2q)$A is odd-normal if ${\mathfrak{S}}_{\left|A\right|}={[1,4pq-1]}_2$. An $m\times n$ integer matrix A is quasi-normal if ${\mathfrak{S}}_A={[1-mn,mn-1]}_2$. An odd-normal BRMR$(p,2q)$A is balanced if A has an equal number of positive and negative numbers in each row and each row sum of $\left|A\right|$ is the same.

To construct new families of trimagic squares of even orders, we need the following lemmas.

In this section, we firstly construct an NMS$(n,3)$ using a trimagic pair of orthogonal DLS$\left(n\right)$s, then based on row-square magic rectangles establish a new product construct theorem, finally provide new families of trimagic squares of even orders by constructing new corresponding infinite classes of row-square magic rectangles.