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Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

The research on orthogonal Steiner systems S (t , k , v) was initiated in 1968. For (t , k) ∈ { (2 , 3) , (3 , 4) } , this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t ≥ 3 and k ≥ 5 , only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S (3 , 5 , v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S (3 , 5 , 4 m + 1) is established for any even m ≥ 2 ; additionally a pair of orthogonal G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is displayed for any odd m ≥ 3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S (5 , 8 , 24) are shown to exist. [ABSTRACT FROM AUTHOR]

Motivated by the application to three-dimensional optical orthogonal codes, we consider the construction for a w -cyclic holey group divisible packing of type (u , w v) with block size three (3-HGDP for short). A maximum w -cyclic 3-HGDP of type (u , w v) contains the largest possible number of base blocks. When u ≡ 0 , 1 (mod 3) , the exact size of maximum w -cyclic 3-HGDP of type (u , w v) has been determined in our previous work. Based on recursive constructions, in this paper we establish a framework to construct maximum w -cyclic 3-HGDPs of type (u , w v) where u ≡ 2 (mod 3). In the process, direct constructions on several key auxiliary designs are displayed by choosing appropriate automorphism groups. Eventually, the sizes of maximum w -cyclic 3-HGDPs of type (u , w v) are determined for all positive integers u , v and w , only leaving a small fraction of possible exceptions unresolved. Furthermore, application of our results to three-dimensional optical orthogonal codes is presented. [ABSTRACT FROM AUTHOR]

This paper concentrates on mutually disjoint 2-chromatic Steiner quadruple systems of order v (briefly by SQS(v)s), meaning that the SQS(v)s are mutually disjoint and they are mutually 2-chromatic having the same two color classes. Following Phelps and Rosa's work, we study maximum sets of mutually disjoint 2-chromatic SQS(v)s by applying direct constructions for suitable collections of triple representatives of V q by utilizing automorphism groups of affine transformations in the finite fields F q for prime powers q. Our main result presents many new maximum sets of mutually disjoint 2-chromatic SQS(v)s. We also improve the lower bounds on the number of mutually disjoint SQS(v)s as a by-product. [ABSTRACT FROM AUTHOR]

In this article, we mainly consider the existence problem of a group divisible design GDD(3,4,n+s) of type 1ns1. We present two recursive constructions for this configuration using candelabra systems and construct explicitly a few small examples admitting given automorphism groups. As an application, several new infinite classes of GDD(3,4,n+s)s of type 1ns1 are produced. Meanwhile a few new infinite families on candelabra quadruple systems with group sizes being odd and stem size greater than one are also obtained. [ABSTRACT FROM AUTHOR]