Abstract A Steiner triple system STS( v ) is called f -pyramidal if it has an automorphism group fixing f points and acting sharply transitively on the remaining $$v-f$$ v - f points. In this paper, we focus on the STSs that are f -pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of f , that is, $$f=0,1,3$$ f = 0 , 1 , 3 . In this paper, we complete this result and determine, for every $$f>3$$ f > 3 , the spectrum of values ( f , v ) for which there is an f -pyramidal STS( v ) over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.

Optical orthogonal codes have applications in optical code-division multiple access communication systems. They can also be used to construct protocol sequences for multiuser collision channel without feedback, and constant weight codes for error detection and correction. Direct constructions with explicit codewords are presented to settle the existence of a J-optimal$(v,4,1)$-optical orthogonal code with$\lfloor (v-1)/12\rfloor $codewords for any positive integer$v\neq 25$. As a corollary, it is shown that an optimal$(v,6,4)$-cyclically permutable constant weight code with$\lfloor (v-1)/12\rfloor $codewords exists for any positive integer$v\neq 25$.

Constant dimension codes (CDCs), as special subspace codes, have received a lot of attention due to their application in random network coding. This paper introduces a family of new codes, called rank metric codes with given ranks (GRMCs), to generalize the parallel construction in [Xu and Chen, IEEE Trans. Inf. Theory, 64 (2018), 6315-6319] and the classic multilevel construction. A Singleton-like upper bound and a lower bound for GRMCs derived from Gabidulin codes are given. Via GRMCs, two effective constructions for CDCs are presented by combining the parallel construction and the multilevel construction. Many CDCs with larger size than the previously best known codes are given. The ratio between the new lower bound and the known upper bound for (4δ, 2δ, 2δ)q-CDCs is calculated. It is greater than 0.99926 for any prime power q and any δ ≥ 3.

Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram rank-metric (FDRM) codes. Via a criterion for linear maximum rank distance (MRD) codes, an explicit construction for a class of systematic MRD codes is presented, which is used to produce new optimal FDRM codes. By exploring the subcodes of Gabidulin codes, if each of the rightmost$\delta -1$columns in the Ferrers diagram$\cal F$has at least$n-r$dots, where$r$is taken in a range, then the conditions that an FDRM code in$\cal F$is optimal are established. The known combining constructions for FDRM code are generalized by introducing the concept of proper combinations of Ferrers diagrams.

In this paper, we consider optimal q-ary cyclic constant-weight codes of length n, minimum distance d, and weight w, briefly cyclic (n, d, w)qcodes. We introduce the pure and mixed difference method to present a combinatorial description for a cyclic (n, d, w)qcode and then obtain some tight upper bounds on the sizes of optimal cyclic (n, d, w)qcodes. Finally, by using Skolem-type sequences, we completely determine the sizes of optimal cyclic (n, d, 3)3codes with minimum distance 1 ≤ d ≤ 6.

Kitayama proposed a novel code-division multiple-access (CDMA) network for image transmission called spatial CDMA. Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial CDMA. An (m,n,k,λ)-OOSPC is a set of m × n (0,1)-matrices with Hamming weight k and maximum correlation value λ. Let Θ (m,n,k,λ) be the largest possible number of codewords among all (m,n,k,λ) -OOSPCs. In this paper, we concentrate on the calculation of the exact value of Θ (m,n,3,1) and the construction of an (m,n,3,1)-OOSPC with Θ (m,n,3,1) codewords. As a consequence, we show that Θ (m,n,3,1)=[ mn-1/6]-1 when mn≡ 14, 20(mod 24), or mn≡ 8, 16(mod 24) and gcd (m,n,4)=2 , or mn≡ 2(mod 6) and gcd (m,n,4)=4 , and Θ (m,n,3,1)= [mn-1/6] otherwise.

In this paper, we study three-dimensional (u × v× w, k, λ) optical orthogonal codes (OOCs) with at most one optical pulse per wavelength/time plane (AM-OPP) restriction, which is denoted by AM-OPP 3-D (u × v × w, k, λ)-OOC. We build an equivalence relation between such an OOC and a certain combinatorial subject, called a w-cyclic group divisible packing of type (vw)u. By this link, the upper bound of the number of codewords is improved and some new combinatorial constructions are presented. As an application, the exact number of codewords of an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC is determined for any positive integers v, w, and u ≠ 2 (mod 6) with some possible exceptions.

Optical orthogonal codes are commonly used as signature codes for optical code-division multiple access systems. So far, research on 2-D optical orthogonal codes has mainly concentrated on the same autocorrelation and cross-correlation constraints. In this paper, we are concerned about optimal 2-D optical orthogonal codes with the autocorrelation λaand the cross-correlation 1. Some combinatorial constructions for 2-D (n×m,k,λa,1) -optical orthogonal codes are presented. Whenk=3 and λa=2, the exact number of codewords of an optimal 2-D (n×m,3,2,1)-optical orthogonal code is determined for any positive integersn≡ 0,1,3,6,9,10 (mod 12) andm≡ 2(mod 4).

Impulse radio sequences (IRSs) were first introduced by Chu and Colbourn in (IEEE Trans. Inf. Theory, Vol. 50, pp. 2402-2407, 2004). Let Ψ(m,k, λ) denote the maximal possible number of sequences in an (m,k, λ)-IRS. In this paper, a fundamental equivalence between impulse radio sequences and strictly cyclic packing having the impulse position property is established. With their relationship, the exact value of Ψ(m,k,k- 1) is finally determined for any positive integersmandk.

In this paper, we are concerned about optimal two-dimensional optical orthogonal codes with λ = 2 . Some combinatorial constructions are presented and many infinite families of optimal two-dimensional optical orthogonal codes with weight 4 and λ = 2 are obtained. Especially, we shall see that in many cases an optimal two-dimensional optical orthogonal code can not achieve the Johnson bound.

Impulse radio sequences were first introduced by Chu and Colbourn in 2004. A better upper bound for(m,k, 1)-IRS was given by Gao and Chang in 2006. In this paper, we present an upper bound for(m,k, lambda)-IRS withlambdages2, which improves the known Johnson bound.

Difference systems of sets (DSSs) are combinatorial structures that are a generalization of cyclic difference sets and arise in connection with code synchronization. In this correspondence, we give some constructions of DSS from cyclic designs and get some infinite classes of optimal difference systems of sets.

Impulse radio sequences (IRSs) were first introduced by Chu and Colbourn in (IEEE Trans. Inf. Theory, Vol. 50, pp. 2402-2407, 2004). In this correspondence, we investigate the upper bound for this class of sequences. By using the relationship between IRSs and CPs, a new upper bound for (m,k,1)-IRS is given, which improves the Johnson bound.

A (v,k,1) optical orthogonal code (OOC), or briefly a (v, k, 1)-OOC, C, is a family of (0,1) sequences of length v and weight k satisfying the following two properties: 1) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/x/sub t+i//spl les/1 for any x=(x/sub 0/x/sub 1/,...,x/sub v-1/)/spl isin/C and any integer i/spl ne/0 (mod v); 2) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/y/sub t+i//spl les/1 for any x=(x/sub 0/x/sub 1/,...,x/sub v-1/)/spl isin/C, y=(y/sub 0/y/sub 1/,...,y/sub v-1/)/spl isin/C with x/spl ne/y, and any integer i, where the subscripts are reduced modulo v. A (v, k,1)-OOC is optimal if it contains /spl lfloor/(v-1)/k(k-1)/spl rfloor/ codewords. In this note, we establish that there exists an optimal (3/sup s/5v, 5,1)-OOC for any nonnegative integer s whenever visa product of primes congruent to 1 modulo 4. This improves the known existence results concerning optimal OOCs.

A (v,k,/spl lambda/) optical orthogonal code C is a family of (0,1) sequences of length v and weight k satisfying the following correlation properties: 1) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/x/sub t+i//spl les//spl lambda/ for any x=(x/sub 0/, x/sub 1/, ..., x/sub v-1/)/spl isin/C and any integer i/spl ne/0(mod v); 2) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/y/sub t+i//spl les//spl lambda/ for any x=(x/sub 0/, x/sub 1/, ..., x/sub v-1/)/spl isin/C, y=(y/sub 0/, y/sub 1/, ..., y/sub v-1/)/spl isin/C with x/spl ne/y, and any integer i, where the subscripts are taken modulo v. A (v,k,/spl lambda/) optical orthogonal code (OOC) with /spl lfloor/(1/k)/spl lfloor/(v-1/k-2)/spl lfloor/(v-2/k-2)/spl lfloor//spl middot//spl middot//spl middot//spl lfloor/(v-/spl lambda//k-/spl lambda/)/spl rfloor/$: M/spl rfloor//spl rfloor//spl rfloor/ codewords is said to be optimal. OOCs are essential for success of fiber-optic code-division multiple-access (CDMA) communication systems. The use of an optimal OOC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, various combinatorial constructions for optimal (v,4,1) OOCs, such as those via skew starters and Weil's theorem on character sums, are given for v/spl equiv/0 (mod 12). These improve the known existence results on optimal OOCs. In particular, it is shown that an optimal (v,4,1) OOC exists for any positive integer v/spl equiv/0 (mod 24).