Abstract
We present unified constructions of optical orthogonalcodes (OOCs) using other combinatorial objects such as cyclic linear codes and frequency hopping sequences.Some of the obtained OOCs are optimal or asymptotically optimal with respect tothe Johnson bound. Also, we are able to show the existence of new optimal frequency hopping sequences (FHSs) with respect to the Singleton bound from our observation on a relation between OOCs and FHSs.The last construction is based on residue rings of polynomials over finite fields, and it yields a new large class of asymptotically optimal $(q-1,k,k-2)$-OOCs for any prime power $q$ with $\gcd{(q-1,k)}=1$. Some infinite families of optimal ones are included as a subclass.
Highlights
An optical orthogonal code is a kind of codes applied in codedivision multiple-access (CDMA) systems in optical fiber networks
We show that the size λ of the intersection of any two distinct blocks from
The following proposition provides a lower bound on the number of blocks of packings of Theorem 3.1 for the case when t = 2 and R = Fq, where q = pn is a prime power
Summary
An optical orthogonal code is a kind of codes applied in codedivision multiple-access (CDMA) systems in optical fiber networks. A (v, k, λa, λc)optical orthogonal code (OOC) is a family of binary sequences (codewords) of length v and constant Hamming weight k satisfying the following two conditions:. We will unify known constructions of (optimal) OOCs using other combinatorial configurations such as cyclic linear codes and frequency hopping sequences. By Remark 1, if we discard all the codewords x for which there exists (i, j) ∈ Zn × Zq−1 \ {(0, 0)} satisfying ωj · σi(x) = x, the resulting packing yields an (n(q − 1), d, n − d)-OOC by taking representatives from block orbits. We apply Theorem 2.2 to two known classes of cyclic linear codes to obtain optimal OOCs. First, we use a special case of “generalized Reed-Solomon codes.”.
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