Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.Comment: 16 page

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

A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page

Accurate detection of moving targets via random sensor arrays and Kerdock codes

The detection and parameter estimation of moving targets is one of the most important tasks in radar. Arrays of randomly distributed antennas have been popular for this purpose for about half a century. Yet, surprisingly little rigorous mathematical theory exists for random arrays that addresses fundamental question such as how many targets can be recovered, at what resolution, at which noise level, and with which algorithm. In a different line of research in radar, mathematicians and engineers have invested significant effort into the design of radar transmission waveforms which satisfy various desirable properties. In this paper we bring these two seemingly unrelated areas together. Using tools from compressive sensing we derive a theoretical framework for the recovery of targets in the azimuth-range-Doppler domain via random antennas arrays. In one manifestation of our theory we use Kerdock codes as transmission waveforms and exploit some of their peculiar properties in our analysis. Our paper provides two main contributions: (i) We derive the first rigorous mathematical theory for the detection of moving targets using random sensor arrays. (ii) The transmitted waveforms satisfy a variety of properties that are very desirable and important from a practical viewpoint. Thus our approach does not just lead to useful theoretical insights, but is also of practical importance. Various extensions of our results are derived and numerical simulations confirming our theory are presented

Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over $\mathbb{Z}_4$. We then specialise to planar monomial functions $f(x)=cx^t$ and present constructions and partial results towards their classification. In particular, we show that $t=1$ is the only odd exponent for which $f(x)=cx^t$ is planar (for some nonzero $c$) over infinitely many fields. The proof techniques involve methods from algebraic geometry.Comment: 23 pages, minor corrections and simplifications compared to the first versio