A Simplified Improvement on the Design of QO-STBC Based on Hadamard Matrices

yesIn this paper, a simplified approach for implementing QO-STBC is presented. It is based on the Hadamard matrix, in which the scheme exploits the Hadamard property to attain full diversity. Hadamard matrix has the characteristic that diagonalizes a quasi-cyclic matrix and decoding matrix that are diagonal matrix permit linear decoding. Using quasi-cyclic matrices in designing QO-STBC systems require that the codes should be rotated to reasonably separate one code from another such that error floor in the design can be minimized. It will be shown that, orthogonalizing the secondary codes and then imposing the Hadamard criteria that the scheme can be well diagonalized. The results of this simplified approach demonstrate full diversity and better performance than the interference-free QO-STBC. Results show about 4 dB gain with respect to the traditional QO-STBC scheme and performs alike with the earlier Hadamard based QO-STBC designed with rotation. These results achieve the consequent mathematical proposition of the Hadamard matrix and its property also shown in this study

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $n\leq 1300$ vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of $n$.Comment: 18 pages, LaTe

Balanced generalized weighing matrices and their applications

Balanced generalized weighing matrices include well-known classical combinatorial objects such as Hadamard matrices and conference matrices; moreover, particular classes of BGW -matrices are equivalent to certain relative difference sets. BGW -matrices admit an interesting geometrical interpretation, and in this context they generalize notions like projective planes admitting a full elation or homology group. After surveying these basic connections, we will focus attention on proper BGW -matrices; thus we will not give any systematic treatment of generalized Hadamard matrices, which are the subject of a large area of research in their own right. In particular, we will discuss what might be called the classical parameter series. Here the nicest examples are closely related to perfect codes and to some classical relative difference sets associated with affine geometries; moreover, the matrices in question can be characterized as the unique (up to equivalence) BGW -matrices for the given parameters with minimum q-rank.One can also obtain a wealth of monomially inequivalent examples and deter  mine the q-ranks of all these matrices by exploiting a connection with linear shift register sequences

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups

Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology. The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6]. In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available. In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code. In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices

(2^n,2^n,2^n,1)-relative difference sets and their representations

We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial f(x)\in \F_{2^n}[x], where $f(x+a)+f(x)+xa$ is a permutation for each nonzero $a$. We call such an $f$ a planar function on \F_{2^n}. The projective plane $\Pi$ obtained from $D$ in the way of Ganley and Spence \cite{ganley_relative_1975} is coordinatized, and we obtain necessary and sufficient conditions of $\Pi$ to be a presemifield plane. We also prove that a function $f$ on \F_{2^n} with exactly two elements in its image set and $f(0)=0$ is planar, if and only if, $f(x+y)=f(x)+f(y)$ for any x,y\in\F_{2^n}