Classes of Optical Orthogonal Codes from Arcs in Root Subspaces T.L. Alders

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Once again, as the coordinates of P G(d, q) are uniquely determined by d + 2 fundamental points we have the following Lemma. Note that in each table, column 1 corresponds to the codes constructed in Section 5. Our construction yields for each λ > 1 an infinite family of OOCs which, in many cases, are asymptotically optimal with respect to the Johnson bound. , cn−1 ) and for any integer 1 ≤ t ≤ n − 1, there holds n−1 X ci ci+t ≤ λa i=0 • (cross-correlation property) for any two distinct codewords c, c0 and for any integer 0 ≤ t ≤ n − 1, there holds n−1 X ci c0i+t ≤ λc i=0 where each subscript is reduced modulo n. 98448 6 Generalizations to P G(d, q k ) For k a positive integer, let Σ = P G(d, q k ). Any two Baer sublines of ` intersect in at most 2 points. If i = 0 then K and φi (K0 ) are in Π and can therefore share as many as d points so λc = d. Identify each arc in F (considered as a point 2d+4 set in Σ) with the corresponding codeword of length q q2 −1−1 and weight q − 1. Subsequently, OOCs have found application for multimedia transmissions in fiber-optic LANs [7]. Denote by Xd the number of NRCs containing P and Q. 12 Tables 2 and 3 show the comparison of the size of some of the codes constructed above with with the Johnson bound. , Constructions of binary constant-weight cyclic codes and cyclically permutable codes, IEEE Trans. Let B1 and B2 be distinct root subspaces of Π containing P . If O` is a full orbit (has size n) then a representative line and corresponding codeword is chosen. Note that both P and Q are extending points of each member of F(B). In one case codewords correspond to (q − 1)-arcs contained in Baer subspaces (and, in general, k th -root subspaces) of a projective space. We start with some basic properties of the geometry of these subspaces of the projective line. Let K1 and K2 be arcs in B1 and B2 respectively, both having P as an extending point. Substituting equation (12), we get q d+1 − 1 q k − 1 (q k(d+1) − q k )(q k(d+1) − q 2k ) · · · (q k(d+1) − q kd ) ZP = B(d, q ) = . Theorem 7 A (d+3)-arc in P G(d, q) is contained in a unique normal rational curve