Symplectic spreads, planar functions and mutually unbiased bases

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $sl_n(\mathbb{C})$ obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra $sl_n(\mathbb{C})$. In the case of symplectic spreads this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.Comment: 20 page

MUBs inequivalence and affine planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes "old" results that do not appear to be well-known concerning known families of complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical Physics 53, 032204 (2012) except for format changes due to the journal's style policie

Interval Semirings

This book has seven chapters. In chapter one we give the basics needed to make this book a self contained one. Chapter two introduces the notion of interval semigroups and interval semifields and are algebraically analysed. Chapter three introduces special types of interval semirings like matrix interval semirings and interval polynomial semirings. Chapter four for the first time introduces the notion of group interval semirings, semigroup interval semirings, loop interval semirings and groupoid interval semirings and these structures are studied. Interval neutrosophic semirings are introduced in chapter five. Applications of these structures are given in chapter six. The final chapter suggests around 120 problems for the reader.Comment: 155 pages; Published by Kappa & Omega in 201