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.

In the first part of this paper [16], some results on how to compute the flat spectra of Boolean constructions w.r.t. the transforms {I, H} n , {H, N } n and {I, H, N } n were presented, and the relevance of Local Complementation to the quadratic case was indicated. In this second part, the results are applied to develop recursive formulae for the numbers of flat spectra of some structural quadratics. Observations are made as to the generalised Bent properties of boolean functions of algebraic degree greater than two, and the number of flat spectra w.r.t. {I, H, N } n are computed for some of them., where i 2 = −1, the Negahadamard kernel, and I the 2 × 2 identity matrix.We say that a Boolean function p(x) : GF(2) n → GF (2) is Bent [17] if P = 2 −n/2 ( n−1 i=0 H)(−1) p(x) has a flat spectrum, or, in other words, if P = (P k ) ∈ C 2 n is such that |P k | = 1 ∀ k ∈ GF(2) n . Bent boolean functions are desirable cryptographic primitives as they optimise resistance to linear cryptanalysis. If the function is quadratic, we can associate to it a simple non-directed graph, and in this case a flat spectrum is obtained iff Γ, the adjacency matrix of the graph, has maximum rank mod 2. In Part I, we generalised C. Riera is with the

In this paper, we analyze the (im)possibility of the exact distinguishability of orthogonal multipartite entangled states under restricted local operation and classical communication. Based on this local distinguishability analysis, we propose a quantum secret sharing scheme (which we call LOCC-QSS). Our LOCC-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 (2,n)-threshold LOCC-QSS scheme, where any two cooperating players, one from each of two disjoint groups of players, can always reconstruct the secret. This LOCC-QSS scheme is quite uncommon, as most (/c,n (-threshold quantum secret sharing schemes have the restriction k > ffl-

Abstract. Bent functions f : F m 2 → 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.

Abstract-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.