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— Generalisations 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… (More)

Generalisations 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… (More)

We relate the one-and two-variable interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions,… (More)

We derive a spectral interpretation of the pivot operation on a graph and generalise this operation to hypergraphs. We establish lower bounds on the number of flat spectra of a Boolean function, depending on internal structures , with respect to the {I, H} n and {I, H, N } n sets of transforms. We also construct a family of Boolean functions of degree… (More)

In the first part of this paper [16], some results on how to compute the flat spectra of Boolean constructions w.r.t. 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.… (More)

We relate the interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions, provide a new… (More)

We describe an operation to dynamically adapt the structure of the Tanner graph used during iterative decoding. Codes on graphs– most importantly, low-density parity-check (LDPC) codes–exploit ran-domness in the structure of the code. Our approach is to introduce a similar degree of controlled randomness into the operation of the message-passing decoder, to… (More)

—In this paper, we propose to enhance the performance of the sum-product algorithm (SPA) by interleaving SPA iterations with a random local graph update rule. This rule is known as edge local complementation (ELC), and has the effect of modifying the Tanner graph while preserving the code. We have previously shown how the ELC operation can be used to… (More)

—We construct Boolean functions whose non-trivial restrictions are either highly nonlinear with respect to the Walsh-Hadamard or the negahadamard transform. We generalise these properties, identify group actions that preserve them, and obtain complementary sets from our functions.

We present a generalised setting for the construction of complementary array pairs and its proof, using a unitary matrix notation. When the unitaries comprise multivariate polynomials in complex space, we show that four definitions of conjugation imply four types of complementary pair-types I, II, III, and IV. We provide a construction for complementary… (More)