Bounds and Constructions of Optimal ($n, 4, 2, 1$) Optical Orthogonal Codes

In this paper, a tight upper bound on the maximum possible code size of n, 4, 2, 1)-OOCs and some direct and recursive constructions of optimal (n, 4, 2, 1)-OOCs attaining the upper bound are given. As consequences, the following new infinite series of optimal (gn,4,2,1)-OOCs are obtained: i) g isin {1,7,11,19,23,31,35,59,71,79,131,179,191,239,251,271,311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971} or g is a prime < 1000 equiv 5 ( mod 8), and n=9<sup>h</sup>25<sup>i</sup>49<sup>i</sup>p<sub>1</sub>p<sub>2</sub>hellipp<sub>r</sub> where h isin {0,1}, i and j are arbitrary nonnegative integers, and each p<sub>i</sub> is a prime equiv 1 ( mod 8); ii) g = 2g' where g' isin {1,7,11,19,23,31,47,71,127,151,167,191,263,271,311,359,367,383,431,439,463,479,503,631,647,719,727,743,823,839,863,887,911,919,967,983,991} and n = p<sub>1</sub>p<sub>2</sub>hellipp<sub>r</sub> where each p<sub>i</sub> is a prime equiv 1 ( mod 4); iii) g isin {4,20} and n is any positive integer prime to 30; iv) g = 8 and n= p<sub>1</sub>p<sub>2</sub>hellipp<sub>r</sub> where each p<sub>i</sub> is a primary equiv 1 ( mod 4) greater than 5.

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