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 (n,4,2,1)-OOCs are obtained: i) $g\in \{1,7,11,19,23,31,35,59,71,79,131,179,191,23,9,251,271, 311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971\}$ or g is a prime < 1000 and $\equiv 5$ (mod 8), and $n=9^h25^i49^jp_1p_2\dots p_r$ where $h \in \{0,1\}$, i and j are arbitrary nonnegative integers, and each $p_i$ is a prime $\equiv 1$ (mod 8); ii) $g = 2g'$ where $g' \in \{1,7,11,19,23,31,47,71,127,151,167,191,263, 271,311,359,367,383,431,439,463,479,503,631,647,719,727,743,823,829,863,887,911,919,967,983,991\}$ and $n=p_1p_2\dots p_r$ where each $p_i$ is a prime $\equiv 1$ (mod 4); iii) $g\in\{4,20\}$ and n is any positive integer prime to 30; iv) g=8 and $n=p_1p_2\dots p_r$ where each $p_i$ is a prime $\equiv1$ (mod 4) greater than 5.

Bounds and constructions of optimal (n,4,2,1) optical orthogonal codes

Abstract

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 (n,4,2,1)-OOCs are obtained: i) $g\in \{1,7,11,19,23,31,35,59,71,79,131,179,191,23,9,251,271, 311,359,379,419,431,439,479,491,499,571,599,631,659,719,739,751,839,971\}$ or g is a prime < 1000 and $\equiv 5$ (mod 8), and $n=9^h25^i49^jp_1p_2\dots p_r$ where $h \in \{0,1\}$, i and j are arbitrary nonnegative integers, and each $p_i$ is a prime $\equiv 1$ (mod 8); ii) $g = 2g'$ where $g' \in \{1,7,11,19,23,31,47,71,127,151,167,191,263, 271,311,359,367,383,431,439,463,479,503,631,647,719,727,743,823,829,863,887,911,919,967,983,991\}$ and $n=p_1p_2\dots p_r$ where each $p_i$ is a prime $\equiv 1$ (mod 4); iii) $g\in\{4,20\}$ and n is any positive integer prime to 30; iv) g=8 and $n=p_1p_2\dots p_r$ where each $p_i$ is a prime $\equiv1$ (mod 4) greater than 5.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/111126
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