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We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence...
In the present paper we consider symmetric Boolean functions with special property. We study properties of the maximal intervals of these functions. Later we show characteristics of corresponding interval graphs and simplified interval graphs. Specifically we prove, that these two graphs are isomorphic for symmetric Boolean function. Then we obtain...
Complete complementary codes (CCCs) is a collection of mutually orthogonal complementary codes (CCs) and Inter-group complementary (IGC) code set consists of multiple disjoint groups of two-dimensional codes. IGC code set has large set size than CCC and both can be applicable in multicarrier code-division multiple access (CDMA). The aperiodic autoc...
We provide a counter example to a conjecture by Leslie Valiant. Most interestingly the counter example was found by introducing guessing numbers - a new graph theoretical concept. We show that solvability of information flow problems of a quite general type is closely related to problems concerning guessing numbers. We reduce a few other conjecture...
This study addresses the observability of Boolean networks (BNs), using semi-tensor product (STP) of matrices. First, unobservable states can be divided into two types, and the first type of unobservable states can be easily determined by blocking idea. Second, it is found that all states reaching to observable states are observable. Based on subgr...
Citations
... The theory of linear approximations, which is based on Hadamard transform of the functions, has been generalized by Danielsen and Parker [3,4] as well as Riera and Parker [19,20], by introducing nega-Hadamard transforms leading to a class of generalized transforms, referred to as HN-transforms, combining Hadamard and nega-Hadamard kernels. It has been observed [19,20] that the quantum error correcting codes with optimal distance appear to have most flat spectra with respect to such transforms. ...
... It has been observed [19,20] that the quantum error correcting codes with optimal distance appear to have most flat spectra with respect to such transforms. In the context of HN-spectra, several results and constructions of Boolean functions and cryptographically strong S-Boxes had been studied in [4,8,16,17,19,21,24]. Surprisingly, while the HNtransform has been used for several purposes, its algorithmic issues have never been studied in detail. ...
\(\textit{HN}\)-transforms, which have been proposed as generalizations of Hadamard transforms, are constructed by tensoring Hadamard and nega-Hadamard kernels in any order. We show that all the \(2^n\) possible \(\textit{HN}\)-spectra of a Boolean function in n variables, each containing \(2^n\) elements (i.e., in total \(2^{2n}\) values in transformed domain) can be computed in \(O(2^{2n})\) time (more specific with little less than \(2^{2n+1}\) arithmetic operations). We propose a generalization of Deutsch-Jozsa algorithm, by employing \(\textit{HN}\)-transforms, which can be used to distinguish different classes of Boolean functions over and above what is possible by the traditional Deutsch-Jozsa algorithm.
... RS codes are widely used for data storage and are suitable for erasure errors (Reed & Solomon, 1960). The second class of error-correcting code that we select is the self-dual additive codes over GF(4) (see Danielsen, 2008 ). These codes can be represented as simple graphs and have many interesting features. ...
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.
... The error-correcting performance of the code depends on the graph state. A graph state itself is sometimes referred as an [[n, 0, d]] self-dual graph code [5]. Second, a special type of graph states, a cluster state, is a universal resource for an one way quantum computer [2,6,7]. ...
... In this work, we investigate a relation between a graph state and a graph code both obtained from the same graph. In some literature, a graph state is also referred as an [[n, k = 0, d]] self-dual graph code [5], but in this work we clearly distinguish both. If logical information can be embedded (k > 0), then it is a graph code. ...
... [12,18], B and Γ(G) are orthogonal, B · Γ(G) = 0. Fig. 3 shows an extended graph from R 5 whose B matrix is (1 1 1 1 1). This graph defines a [ [5,1]] graph code that performs an encoding by spreading 1-qubit logical information in the input vertex 0 over all the output vertices. ...
A graph state and a graph code respectively are defined based on a
mathematical simple graph. In this work, we examine a relation between a graph
state and a graph code both obtained from the same graph, and show that a graph
state is a superposition of logical qubits of the related graph code. By using
the relation, we first discuss that a local complementation which has been used
for a graph state can be useful for searching locally equivalent stabilizer
codes, and second provide a method to find a stabilizer group of a graph code.
