Matthew G. Parker

Optimal sequences for channel estimation using discrete Fourier transform techniques

This paper addresses the problem of selecting the optimum training sequence for channel estimation in communication systems over time-dispersive channels. By processing in the frequency domain, a new explicit form of search criterion is found, the gain loss factor (GLF), which minimizes the variance of the estimation error and is easy to compute. Theoretical upper and lower bounds on the GLF are derived. An efficient directed search strategy and optimal sequences up to length 42 are given. These sequences are optimal only for frequency domain estimation, not for time domain estimation.

Generalized Bent Criteria for Boolean Functions (I)

Generalizations of the bent property of a Boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic Boolean functions are related to simple graphs and it is shown that the orbit generated by successive local complementations on a graph can be found within the transform spectra under investigation. The flat spectra of a quadratic Boolean function are related to modified versions of its associated adjacency matrix

On the classification of all self-dual additive codes over GF(4) of length up to 12

We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively.

Sequences and Their Applications - SETA 2008

The purpose of this paper is to provide a brief survey of CCZ and EA equivalence for functions f : G → N where G and N are finite and N is abelian, and, for the case f : Zp → Zp , to investigate two codes derived from f , inspired by these equivalences. In particular we show the dimension of the kernel of each code determines a new invariant of the corresponding equivalence class. We present computational results for p = 2 and small m.

On Boolean functions which are bent and negabent

Bent functions f : F2m → F2 achieve largest distance to all linear functions. Equivalently, their spectrum with respect to the Hadamard-Walsh transform is flat (i.e. all spectral values have the same absolute value). That is equivalent to saying that the function f has optimum periodic autocorrelation properties. Negaperiodic correlation properties of f are related to another unitary transform called the nega-Hadamard transform. A function is called negabent if the spectrum under the nega-Hadamard transform is flat. In this paper, we consider functions f which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties. Several constructions and classifications are presented.

A multi-dimensional approach to the construction and enumeration of Golay complementary sequences

We argue that a Golay complementary sequence is naturally viewed as a projection of a multi-dimensional Golay array. We present a three-stage process for constructing and enumerating Golay array and sequence pairs:1.construct suitable Golay array pairs from lower-dimensional Golay array pairs; 2.apply transformations to these Golay array pairs to generate a larger set of Golay array pairs; and 3.take projections of the resulting Golay array pairs to lower dimensions. This process greatly simplifies previous approaches, by separating the construction of Golay arrays from the enumeration of all possible projections of these arrays to lower dimensions. We use this process to construct and enumerate all 2^h-phase Golay sequences of length 2^m obtainable under any known method, including all 4-phase Golay sequences obtainable from the length 16 examples given in 2005 by Li and Chu [Y. Li, W.B. Chu, More Golay sequences, IEEE Trans. Inform. Theory 51 (2005) 1141-1145].

Self-dual bent functions

A bent function is called self-dual if it is equal to its dual. It is called anti-self-dual if it is equal to the complement of its dual. A spectral characterisation in terms of the Rayleigh quotient of the Sylvester Hadamard matrix is derived. Bounds on the Rayleigh quotient are given for Boolean functions in an odd number of variables. An efficient search algorithm based on the spectrum of the Sylvester matrix is derived. Primary and secondary constructions are given. All self-dual bent Boolean functions in ≤ 6 variables and all quadratic such functions in eight variables are given, up to a restricted form of affine equivalence.

Quantum secret sharing based on local distinguishability

In this paper we analyze the (im)possibility of the exact distinguishability of orthogonal multipartite entangled states under {\em restricted local operation and classical communication}. Based on this local distinguishability analysis we propose a new scheme for quantum secret sharing (QSS). Our QSS scheme is quite general and cost efficient compared to other schemes. In our scheme no joint quantum operation is needed to reconstruct the secret. We also present an interesting

Negabent Functions in the Maiorana---McFarland Class

(2,n)

Generalised Rudin-Shapiro Constructions

-threshold QSS scheme, where any two cooperating players, one from each of two disjoint groups of players, can always reconstruct the secret. This QSS scheme is quite uncommon, as most