Constanza Riera

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

One and two-variable interlace polynomials: a spectral interpretation

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, provide new one and two-variable interlace polynomials, and propose a generalisation of the interlace polynomial to hypergraphs. We also prove conjectures from [15] and equate certain spectral metrics with various evaluations of the interlace polynomial.

Iterative decoding on multiple tanner graphs using random edge local complementation

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 implement an iterative permutation group decoder (SPA-PD)-one of the most successful iterative soft-decision decoding strategies at small blocklengths. In this work, we exploit the fact that ELC can also give structurally distinct parity-check matrices for the same code. Our aim is to describe a simple iterative decoder, running SPA-PD on distinct structures, based entirely on random usage of the ELC operation. This is called SPA-ELC, and we focus on small blocklength codes with strong algebraic structure. In particular, we look at the extended Golay code and two extended quadratic residue codes. Both error rate performance and average decoding complexity, measured by the average total number of messages required in the decoding, significantly outperform those of the standard SPA, and compares well with SPA-PD. However, in contrast to SPA-PD, which requires a global action on the Tanner graph, we obtain a performance improvement via local action alone. Such localized algorithms are of mathematical interest in their own right, but are also suited to parallel/distributed realizations.

Generalised complementary arrays

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 pairs of types I, II, and III over {1,−1}, and further specialize to a construction for all known 2 ×2 ×…×2 complementary array pairs of types I, II, and III over {1,−1}. We present a construction for type-IV complementary array pairs, and call them Rayleigh quotient pairs . We then generalise to complementary array sets, provide a construction for complementary sets of types I, II, and III over {1,−1}, further specialize to a construction for all known 2 ×2 ×…×2 complementary array sets of types I, II, and III over {1,−1}, and derive closed-form Boolean formulas for these cases.

Adaptive Soft-Decision Iterative Decoding Using Edge Local Complementation

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 randomness 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 improve the performance of iterative decoding of classical structured (i.e., non-random) codes for which strong code properties are known. We use ideas similar to Halford and Chugg (IEEE Trans. on Commun., April 2008), where permutations on the columns of the parity-check matrix are drawn from the automorphism group of the code, Aut

From graph states to two-graph states

\mathcal{(C)}

Boolean functions whose restrictions are highly nonlinear

. The main contributions of our work are: 1) We maintain a graph-local perspective, which not only gives a low-complexity, distributed implementation, but also suggests novel applications of our work, and 2) we present an operation to draw from Aut

A Complete Characterization of Plateaued Boolean Functions in Terms of Their Cayley Graphs

\mathcal{(C)}

Random Edge-Local Complementation with Applications to Iterative Decoding of High-Density Parity-Check Codes

such that graph isomorphism is preserved, which maintains desirable properties while the graph is being updated. We present simulation results for the additive white Gaussian noise (AWGN) channel, which show an improvement over standard sum-product algorithm (SPA) decoding.

On Pivot Orbits of Boolean Functions

The name ‘graph state’ is used to describe a certain class of pure quantum state which models a physical structure on which one can perform measurement-based quantum computing, and which has a natural graphical description. We present the two-graph state, this being a generalisation of the graph state and a two-graph representation of a stabilizer state. Mathematically, the two-graph state can be viewed as a simultaneous generalisation of a binary linear code and quadratic Boolean function. It describes precisely the coefficients of the pure quantum state vector resulting from the action of a member of the local Clifford group on a graph state, and comprises a graph which encodes the magnitude properties of the state, and a graph encoding its phase properties. This description facilitates a computationally efficient spectral analysis of the graph state with respect to operations from the local Clifford group on the state, as all operations can be realised graphically. By focusing on the so-called local transform group, which is a size 3 cyclic subgroup of the local Clifford group over one qubit, and over n qubits is of size 3n, we can efficiently compute spectral properties of the graph state.