Abstract In this paper, we investigate a class of optimal circulant { 0 , 1 } -arrays other than the previously known class of optimal designs for fMRI experiments with a single type of stimulus. We suppose throughout the paper that n ≡ 2 ( mod 4 ) and discuss the asymptotic optimality and the D-efficiency of k × n circulant almost orthogonal arrays (CAOAs) with 2 levels (presence/absence of the stimulus), strength 2 and bandwidth 1, denoted by CAOA ( n , k , 2 , 2 , 1 ) . We show that for n ≡ 2 ( mod 4 ) the largest possible value of k for statistically optimal CAOA ( n , k , 2 , 2 , 1 ) cannot exceed n ∕ 2 . We also clarify that CAOA ( n , k , 2 , 2 , 1 ) with high D-efficiency and k greater than n ∕ 2 can be obtained via perfect binary sequences. By applying algebraic constructions for perfect binary sequences and by computer search, lists of such efficient CAOAs and the new class of optimal CAOAs are provided.

中文翻译：

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通过两级循环几乎正交阵列进行 fMRI 实验的优化和有效设计

摘要 在本文中，我们研究了一类最佳循环 { 0 , 1 } 阵列，而不是先前已知的一类用于单一类型刺激的 fMRI 实验的最佳设计。我们在整篇论文中假设 n ≡ 2 ( mod 4 ) 并讨论具有 2 个级别（存在/不存在刺激）、强度 2 和带宽的 k × n 循环几乎正交阵列（CAOA）的渐近最优性和 D 效率1，由 CAOA ( n , k , 2 , 2 , 1 ) 表示。我们表明，对于 n ≡ 2 (mod 4)，统计上最优 CAOA (n, k, 2, 2, 1) 的最大可能 k 值不能超过 n ∕ 2 。我们还阐明，可以通过完美的二进制序列获得具有高 D 效率且 k 大于 n ∕ 2 的 CAOA ( n , k , 2 , 2 , 1 )。通过对完美二进制序列应用代数构造和通过计算机搜索，