Let N={n0, n1, ... , nk-1} be a set of positive integers and M= {m0, m1, ... , mk-1} be a multiset of positive integers. By an (N, M, w,1; λ)-multilength optical orthogonal code (MLOOC), we mean an MLOOC of autocross correlation value and intracross correlation value one and intercross correlation value λ. The code contains micodewords of weight w and length nifor 0 ≤ i ≤ k-1. The study of MLOOCs is motivated by an application in optical networks requiring multiple signaling rates and quality-of-services. In this paper, we study (N, M, w,1; λ)-MLOOCs with λ =2 (the least value among the nontrivial intercross correlations). Some new upper bounds on code size are derived under certain restrictions and a novel encoding approach is established. A number of series of new MLOOCs are then produced. These codes are of optimal sizes with respect to the new bounds.

We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known q-ary MDS codes have length O(q), we show that q-ary MDS symbol-pair codes can have length Ω(q2). In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters.

Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise to optimal frequency-hopping sequences and optimal comma-free codes. New constructions for PCDPs, based on almost difference sets and cyclic difference matrices, are given. These produce new infinite families of optimal PCDPs (and hence optimal frequency-hopping sequences and optimal comma-free codes). The existence problem for optimal PCDPs in BBZ3m, withmbase blocks of size three, is also solved for allm≠ 8,16 mod 24.

A(nxm,k, ¿) two-dimensional optical orthogonal code (2-D OOC),C, is a family ofnxm(0, 1)-arrays of constant weightksuch that¿i=1n¿j=0m-1A(i,j)B(i,j¿m¿) ¿ ¿ for any arraysA,BinCand any integer¿except whenA=Band¿ ¿ 0(modm), where¿mdenotes addition modulom. Such codes are of current practical interest as they enable optical communication at lower chip rate. To simplify practical implementation, the AM-OPPW (at most one-pulse per wavelength) restriction is often appended to a 2-D OOC. An AM-OPPW 2-D OOC is optimal if its size is the largest possible. In this paper, the notion of a perfect AM-OPPW 2-D OOC is proposed, which is an optimal(nxm,k, ¿) AM-OPPW 2-D OOC with cardinality[(m¿n(n-1)...(n-¿))/(k(k-1)...(k-¿))] . A link between optimal(nxm,k, ¿) AM-OPPW 2-D OOCs and block designs is developed. Some new constructions for such optimal codes are described by means of semicyclic group divisible designs. Several new infinite families of perfect(nxm,k, 1) AM-OPPW 2-D OOCs withk¿ {2, 3, 4} are thus produced.

In communication systems, frequency-hopping spread spectrum and direct-sequence spread spectrum are two main spread coding technologies. Frequency-hopping sequences are needed in FH-CDMA systems. In this correspondence, three algebraic constructions of sets of optimal frequency-hopping sequences are presented. The parameters of these sets of frequency-hopping sequences are new and flexible.

Signal sets with the best correlation property are desirable in code-division multiple-access (CDMA) systems. In this paper, the construction of Wootters and Fields for mutually unbiased bases is extended into a generic construction of signal sets using planar functions. Then, specific classes of planar functions and almost bent functions are employed to obtain (q2+q,q) signal sets. The signal sets derived from planar functions are optimal with respect to the Levenstein bound, and those obtained from almost bent functions nearly meet the Levenstein bound. The signal sets constructed in this paper could have a very small alphabet size, and have applications in synchronous DS-CDMA systems, where the number of users is greater than the signal space dimension or the spreading factor

There are two kinds of perfect (k-t)-deletion-correcting codes with words of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(t,k,v)-codes and T(t,k,v)-codes respectively. Both a T*(t,k,v)-code and a T(t,k,v)-code are capable of correcting any combination of up to (k-t) deletions and insertions of letters occurred in transmission of codewords. In this correspondence, we consider constructions for the codes from directed designs. By means of these constructions, the existence of a T(2,7,v)-code is settled for all positive integers v with the exception of 68 values of v; T*(2,7,v)-codes are constructed for all integers vges2350. A large number of explicit constructions for T*(2,7,v)-codes with v<2350 are also presented

Constant-composition codes are a special class of constant-weight codes. They include permutation codes as a subclass. In this correspondence, two classes of optimal constant-composition codes are constructed. Both constructions employ perfect nonlinear functions, but they are different.

Constant-composition codes (CCCs) are a special class of constant-weight codes. They include permutation codes as a subclass. In this correspondence, a link between CCCs and generalized double resolvable packing designs is developed, and used to construct several infinite series of optimal CCCs.

Optimal (9p,4,1) optical orthogonal codes (OOCs) are constructed for all primes p congruent to 1 modulo 4. Direct constructions with explicit codewords are presented for the case $p \equiv 13 \ \mbox{mod} \ 24$, and Weil's theorem on character sums is used to settle the cases $p \equiv 1,5,17 \ \mbox{mod} \ 24$. By applying a known recursive construction, optimal (9v,4,1)-OOCs are obtained for all v, a product of primes congruent to 1 modulo 4.

Four direct constructions, three of which are by way of skew starters, are given in this correspondence for optimal (/spl upsi/, 4, 1) optical orthogonal codes (OOC's). These improve the known existence results concerning optimal (/spl upsi/, 4, 1)-OOCs. In particular, it is shown that an optimal (/spl upsi/, 4, 1)-OOC exists for all positive integers /spl upsi//spl equiv/6 (mod 12) or /spl upsi//spl equiv/24 (mod 48). It is also shown that an optimal (12/spl upsi/, 4, 1)-OOC exists for any positive integer /spl upsi/ whose prime factors are all congruent to 1 modulo 4.