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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]

Abstract The combinatorial object named t -spontaneous emission error design (t -SEED) was proposed by Beth et al. in 2003 in order to correct errors caused by quantum jumps. The newly rising category of t -SEEDs has been studied extensively in recent years. Especially, the maximal possible dimensions for 2-SEEDs with block size 3 were determined completely; lower bounds on 2-SEEDs were established by applying affine groups. In this paper we utilize the action of twisted affine groups on finite fields and obtain new lower bounds on the dimensions of 2- (q 2 , k ; m) SEEDs, some of which outperform the known ones. [ABSTRACT FROM AUTHOR]

A t-spontaneous emission error design, denoted by t- [ABSTRACT FROM AUTHOR]

Quantum jump codes are quantum error-correcting codes which correct errors caused by quantum jumps. A t-spontaneous emission error design ( t-SEED) was introduced by Beth et al. in 2003 [T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, A new class of designs which protect against quantum jumps, Des Codes Cryptogr 29 (2003), 51-70.] to construct quantum jump codes. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. A nondegenerate t-SEED is optimal if it has the largest possible dimension. In this paper, we investigate the bounds on the dimensions of 2-SEEDs systematically. The exact dimensions of optimal 2- [ABSTRACT FROM AUTHOR]