Associative memories enjoy many interesting properties in terms of error correction capabilities, robustness to noise, storage capacity, and retrieval performance, and their usage spans over a large set of applications. In this letter, we investigate and extend tournament-based neural networks, originally proposed by Jiang, Gripon, Berrou, and Rabbat (2016), a novel sequence storage associative memory architecture with high memory efficiency and accurate sequence retrieval. We propose a more general method for learning the sequences, which we call feedback tournament-based neural networks. The retrieval process is also extended to both directions: forward and backward-in other words, any large-enough segment of a sequence can produce the whole sequence. Furthermore, two retrieval algorithms, cache-winner and explore-winner, are introduced to increase the retrieval performance. Through simulation results, we shed light on the strengths and weaknesses of each algorithm.

Clique-based neural associative memories introduced by Gripon and Berrou (GB), have been shown to have good performance, and in our previous work we improved the learning capacity and retrieval rate by local coding and precoding in the presence of partial erasures. We now take a step forward and consider nested-clique graph structures for the network. The GB model stores patterns as small cliques, and we here replace these by nested cliques. Simulation results show that the nested-clique structure enhances the clique-based model.

Techniques from coding theory are able to improve the efficiency of neuroinspired and neural associative memories by forcing some construction and constraints on the network. In this letter, the approach is to embed coding techniques into neural associative memory in order to increase their performance in the presence of partial erasures. The motivation comes from recent work by Gripon, Berrou, and coauthors, which revisited Willshaw networks and presented a neural network with interacting neurons that partitioned into clusters. The model introduced stores patterns as small-size cliques that can be retrieved in spite of partial error. We focus on improving the success of retrieval by applying two techniques: doing a local coding in each cluster and then applying a precoding step. We use a slightly different decoding scheme, which is appropriate for partial erasures and converges faster. Although the ideas of local coding and precoding are not new, the way we apply them is different. Simulations show an increase in the pattern retrieval capacity for both techniques. Moreover, we use self-dual additive codes over field [Formula: see text], which have very interesting properties and a simple-graph representation.

In this paper, some questions on the distribution of the peak-to-mean envelope power ratio (PMEPR) of standard binary Golay sequences are solved. For n odd, we prove that the PMEPR of each standard binary Golay sequence of length 2nis exactly 2, and determine the location(s), where peaks occur for each sequence. For n even, we prove that the envelope power of such sequences can never reach 2n+1at time points t ∈ {(v/2u)|0 ≤ v ≤ 2u, v,u ∈ N}. We further identify eight sequences of length 24and eight sequences of length 26that have PMEPR exactly 2, and raise the question whether, asymptotically, it is possible for standard binary Golay sequences to have PMEPR less than 2 - ϵ, where, ϵ > 0.

We describe the application of edge-local complementation (ELC) to a Tanner graph associated with a binary linear code, C. Various properties of ELC are described, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), and the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upper-bounded. ELC generates all systematic parity-check matrices (the orbit) of the code, so WB-ELC facilitates a restriction to low-weight matrices of this orbit. We propose using ELC and WB-ELC as a source of diversity, to improve iterative soft-input soft-output decoding of high-density parity-check (HDPC) codes, with the sum-product algorithm (SPA). A motivation of ELC-enhanced SPA decoding is locality; that diversity is achieved by local graph action, and is well-suited to the local actions that constitute the SPA and allows for parallel software implementation. Simulation data on the error-rate performance of the proposed SPA-ELC and SPA-WBELC iterative decoding algorithms is shown for several HDPC codes. A gain is reported over SPA decoding, and over a recently proposed algorithm to decode HDPC codes using permutations from Aut(C). ELC-enhanced decoding extends the scope of iterative decoding to codes with trivial Aut(C).

We show that (n, 2 n ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 n ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.

This work is an extension of our previous work on an iterative soft-decision decoder for high-density parity-check codes, using a graph-local operation known as edge-local complementation (ELC). Inferred least reliable codeword positions are targeted by an ELC stage in between sum-product algorithm iterations. A gain is shown over related iterative decoding algorithms - mainly due to an improved heuristic to determine optimum ELC locations in the Tanner graph - both in error-rate performance, as well as complexity in terms of a significant reduction in the required number of ELC operations. We also present a novel damping operation, generalized to the graph-local setting where extrinsic information remains on edges not affected by ELC.

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.

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.

In 1999, Davis and Jedwab gave an explicit algebraic normal form for$m! \cdot 2^{h(m+2)}$ordered Golay pairs of length$2^{m}$over$ {\BBZ }_{2^{h}}$, involving$m!/2 \cdot 2^{h(m+1)}$Golay sequences. In 2005, Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a “crossover” of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length$2^{m}$for$m > 4$under BudiŠin's 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn's 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length$2^{m}$over$ {\BBZ }_{2^{h}}$can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length$2^{m}$are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of$2$, and to alphabets other than$ {\BBZ }_{2^{h}}$.

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.

A new metric, the multivariate merit factor (MMF) of a Boolean function, is presented, and various infinite recursive quadratic sequence constructions are given for which both univariate and multivariate merit factors can be computed exactly. In some cases these constructions lead to merit factors with non-vanishing asymptotes. A formula for the average value of 1/MMF is derived and a characterisation of the MMF in terms of cryptographic differentials is discussed.

A method is described for the construction of binary sequences of very long length and with asymptotic merit factor>6.3. The result is backed up by strong experimental evidence although no formal proof for the asymptote is known. The sequences were found by Kristiansen using a small deterministic search, which we describe. Borwein, Choi, and Jedwab have independently identified a merit factor asymptote of 6.3421.... After we became aware of their work we realized that the sequences we construct are more simply described as periodic extensions of periodically rotated Legendre sequences.

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.