No abstract

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called Alltop functions. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.

Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in C d are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in C d if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev.A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350-354; Wootters and Fields 1989 Ann. Phys. 191 363-381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19-L20, 2004) that the existence of a complete set of MUBs in C d is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case.