We consider the problem of exhaustively visiting all pairs of linear cellular automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of coprime polynomials over a finite field having the same degree and a nonzero constant term. While previous research showed how to count all such pairs for a given degree and order of the finite field, no practical enumeration algorithms have been proposed so far. Here, we start closing this gap by addressing the case of polynomials defined over the field $\F_2$, which corresponds to binary CA. In particular, we exploit Benjamin and Bennett's bijection between coprime and non-coprime pairs of polynomials, which enables us to organize our study along three subproblems, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. In the course of this investigation, we unveil interesting connections with algebraic language theory and combinatorics, obtaining an enumeration algorithm and an alternative derivation of the counting formula for this problem.

翻译：我们考虑了穷举访问所有能产生正交拉丁方的线性元胞自动机对的问题，即线性正交元胞自动机对。该问题等价于枚举有限域上所有具有相同次数和非零常数项的多项式互质对。尽管先前研究已表明如何对给定次数和有限域阶数计数所有此类对，但目前尚未提出实用的枚举算法。本文通过针对定义在域$\F_2$上的多项式（对应二元元胞自动机）的情况，开始填补这一空白。特别地，我们利用Benjamin与Bennett关于互质多项式对与非互质多项式对之间的双射关系，将研究组织为三个子问题：（1）常数项序列的枚举与计数，（2）次数序列的枚举与计数，（3）中间项序列的枚举与计数。在此过程中，我们揭示了与代数语言理论和组合学之间的有趣关联，并获得了该问题的枚举算法及计数公式的另一种推导。