Abstract
A set S of permutations of k objects is μ-uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly μ permutations in S mapping A onto B. Arithmetical conditions and symmetries are discussed. We describe the character-theoretic method which is useful if S is contained in a permutation group. A main result is the construction of a 2-uniform, 2-homogeneous set of permutations on 6 objects and of a 3-uniform, 3-homogeneous set of permutations on 9 objects. These are contained in the simple permutation groups PSL 2(5) and PSL 2(8), respectively. The result is useful in the framework of theoretical secrecy and authentication (see Stinson 1990, Bierbrauer and Tran 1991).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beth, T. Jungnickel, D. and Lenz, H., Design Theory, Mannheim-Wien-Zürich: Bibl. Institut, 1985.
Bierbrauer, J. and van Trung, Tran, Halving PGL(2, 2% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttL% earyqr1ngBPrgaiuaacqWFsgWGaaa!3FBE!\[f\]), % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttL% earyqr1ngBPrgaiuaacqWFsgWGaaa!3FBE!\[f\] odd: a series of cryptocodes, Designs, Codes and Cryptography, 1: 141–148, 1991.
Cameron, Peter J., 1988. Geometric sets of permutations, Geometriae Dedicata 25, 339–351.
Kantor, W.M. 1972. On incidence matrices of finite projective planes and affine spaces, Mathematische Zeitschrift 124: 315–318.
O'Nan, M.E. 1985. Sharply 2-transitive sets of permutations, Proc. Rutgers Group Theory Year 1983–1984 (ed. M. Aschbacher et. al), Cambridge: Cambridge Universities Press.
Stinson, D.R. 1990. The combinatorics of authentication and secrecy codes, Journal of Cryptology 2: 23–49.
Stinson, D.R. and Teirlinck, Luc., 1990. A construction for authentication/secrecy codes from 3-homogeneous permutation groups, European Journal of Combinatatorics 11: 73–79.
Witt, E. 1938. Über Steinersche Systems, Abhandlungen des Mathematischen Seminars Hamburg 12: 265–275.
Author information
Authors and Affiliations
Additional information
Communicated by D. Jungnickel
Rights and permissions
About this article
Cite this article
Bierbrauer, J., Van Trung, T. Some highly symmetric authentication perpendicular arrays. Des Codes Crypt 1, 307–319 (1991). https://doi.org/10.1007/BF00124606
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00124606