A weak difference set construction for higher dimensional designs
Designs, Codes and Cryptography
Cocyclic Development of Designs
Journal of Algebraic Combinatorics: An International Journal
Finite fields
A unifying construction for difference sets
Journal of Combinatorial Theory Series A
Cocyclic Generalised Hadamard Matrices and Central RelativeDifference Sets
Designs, Codes and Cryptography
Translates of linear codes over Z4
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Relative difference sets fixed by inversion (ii): character theoretical approach
Journal of Combinatorial Theory Series A
Bundles, presemifields and nonlinear functions
Designs, Codes and Cryptography
Differentially 2-uniform cocycles: the binary case
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Equivalence classes of multiplicative central (pn, pn, pn, 1)-relative difference sets
Cryptography and Communications
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Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (pa, pa, pa, 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (pa, pa, pa, 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (pa, pa, pa, 1)-RDS, and give a second new construction using presemifields.We use this cohomological approach to identify equivalence classes of central (pa, pa, pa, 1)-RDS with elementary abelian C and E/C. We show that for p = 2, a ≤ 3 and p = 3, a ≤ 2, every central (pa, pa, pa, 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p = 2, a = 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (pa, pa, pa, 1)-RDS, of which one is abelian and a − 1 are non-abelian.