Examples Of Such Codes Research Articles

Weight distribution of low-density periodic random errors and their correcting codes with error decoding probability

We present weight distribution of low-density periodic random errors in the space of all q-ary n-tuples along with the average Hamming weight of the error set. We also provide necessary and sufficient conditions for the existence of linear codes correcting such error pattern. Examples of such codes are given. Finally, probability of decoding error of such codes over a binary symmetric channel is derived.

Designing Codes around Interactions: The Case of a Spin.

I present a new approach for designing quantum error-correcting codes guaranteeing a physically natural implementation of Clifford operations. Inspired by the scheme put forward by Gottesman, Kitaev, and Preskill for encoding a qubit in an oscillator in which Clifford operations may be performed via Gaussian unitaries, this approach yields new schemes for encoding a qubit in a large spin in which single-qubit Clifford operations may be performed via spatial rotations. I construct all possible examples of such codes, provide universal-gate-set implementations using quadratic angular-momentum Hamiltonians, and derive criteria for when these codes exactly correct physically relevant errors.

Quasi-Cyclic LDPC Codes With Parity-Check Matrices of Column Weight Two or More for Correcting Phased Bursts of Erasures

In his pioneering work on LDPC codes, Gallager dismissed codes with parity-check matrices of weight two after proving that their minimum Hamming distances grow at most logarithmically with their code lengths. In spite of their poor minimum Hamming distances, it is shown that quasi-cyclic LDPC codes with parity-check matrices of column weight two have good capability to correct phased bursts of erasures which may not be surpassed by using quasi-cyclic LDPC codes with parity-check matrices of column weight three or more. By modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. Quasi-cyclic LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel are also considered. Examples of such codes based on Reed-Solomon and Gabidulin codes are presented.

ПИРАМИДАЛЬНАЯ СХЕМА ПОСТРОЕНИЯ БИОРТОГОНАЛЬНЫХ ВЕЙВЛЕТ-КОДОВ НАД КОНЕЧНЫМИ ПОЛЯМИ

The existence of a biorthogonal decomposition of the space V of dimension n over the field GF(q) is constructively proved, namely, two representations of it are obtained as direct sums of subspaces V = W0⊕W1⊕. . .⊕WJ⊕VJ and V = W˜0⊕W˜1⊕. . .⊕W˜J⊕V˜J ,such that at the j-th level of the decomposition, for 0 < j 6 J, Vj−1 = Vj⊕Wj , V˜j−1 == V˜j ⊕ W˜j , the subspace Vj is orthogonal to W˜j , and the subspace Wj is orthogonal to V˜j . The partition of the space at the j-th level is made with the help of pairs of level filters (hj, gj) and (h˜j, g˜j), for the construction of which the corresponding algorithms have been developed and theoretically proved. A new family of biorthogonal wavelet codes is built on the basis of the multilevel wavelet decomposition scheme with coding rate 2−L, where L is the number of used decomposition levels, and examples of such codes are given.

Some codes in symmetric and linear groups

For a finite group G, a positive integer λ, and subsets X,Y of G, write λG=XY if the products xy (x∈X,y∈Y), cover G precisely λ times. Such a subset Y is called a code with respect to X, and when λ=1 it is a perfect code in the Cayley graph Cay (G, X). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups SL2(q). We also propose conjectures about the existence of some much wider families.

Codes detecting, locating and correcting random errors occurring in multiple sub-blocks

In this paper, we study bounds on the number of check digits of linear codes that can detect multiple sub-blocks each affected by e or less random errors and can locate such corrupted multiple sub-blocks. Further, we obtain an upper bound on the number of check digits of linear codes which can correct such errors occurring in multiple sub-blocks. We also give examples of such codes.

On the number of inequivalent Gabidulin codes

Maximum rank-distance (MRD) codes are extremal codes in the space of $m\times n$ matrices over a finite field, equipped with the rank metric. Up to generalizations, the classical examples of such codes were constructed in the 1970s and are today known as Gabidulin codes. Motivated by several recent approaches to construct MRD codes that are inequivalent to Gabidulin codes, we study the equivalence issue for Gabidulin codes themselves. This shows in particular that the family of Gabidulin codes already contains a huge subset of MRD codes that are pairwise inequivalent, provided that $2\le m\le n-2$.

DNA cyclic codes over rings

In this paper we construct new DNA cyclic codes over rings. Firstly, we introduce a new family of DNA cyclic codes over the ring $R=\mathbb{F}_2[u]/(u.6)$. A direct link between the elements of such a ring and the $64$ codons used in the amino acids of the living organisms is established. Using this correspondence we study the reverse-complement properties of our codes. We use the edit distance between the codewords which is an important combinatorial notion for the DNA strands. Next, we define the Lee weight, the Gray map over the ring $R$ as well as the binary image of the DNA cyclic codes allowing the transfer of studying DNA codes into studying binary codes. Secondly, we introduce another new family of DNA skew cyclic codes constructed over the ring $\tilde {R}=\mathbb{F}_2+v\mathbb{F}_2=\{0, 1, v, v+1\}, $ where $v^2=v$. The codes obtained are cyclic reverse-complement over the ring $\tilde {R}$. Further we find their binary images and construct some explicit examples of such codes.

A new theory of development: the generation of complexity in ontogenesis.

Today there is a very wide consensus on the idea that embryonic development is the result of a genetic programme and of epigenetic processes. Many models have been proposed in this theoretical framework to account for the various aspects of development, and virtually all of them have one thing in common: they do not acknowledge the presence of organic codes (codes between organic molecules) in ontogenesis. Here it is argued instead that embryonic development is a convergent increase in complexity that necessarily requires organic codes and organic memories, and a few examples of such codes are described. This is the code theory of development, a theory that was originally inspired by an algorithm that is capable of reconstructing structures from incomplete information, an algorithm that here is briefly summarized because it makes it intuitively appealing how a convergent increase in complexity can be achieved. The main thesis of the new theory is that the presence of organic codes in ontogenesis is not only a theoretical necessity but, first and foremost, an idea that can be tested and that has already been found to be in agreement with the evidence.

Algebraic space-time codes based on division algebras with a unitary involution

In this paper, we focus on the design of unitary space-time codes achieving full diversity using division algebras, and on the systematic computation of their minimum determinant.We also give examples of such codes with high minimum determinant. Division algebras allow to obtain higher rates than known constructions based on finite groups.

Quantum error correction via convex optimization

We show that the problem of designing a quantum information error correcting procedure can be cast as a bi-convex optimization problem, iterating between encoding and recovery, each being a semidefinite program. For a given encoding operator the problem is convex in the recovery operator. For a given method of recovery, the problem is convex in the encoding scheme. This allows us to derive new codes that are locally optimal. We present examples of such codes that can handle errors which are too strong for codes derived by analogy to classical error correction techniques.

Preface

Preface

Linear Decoupled and Quasi-Decoupled Space–Time Codes

Space-time transmit diversity results in coupling of transmitted symbols across different antennas, which increases the complexity of maximum-likelihood decoding. Symbol coupling can be completely or partially avoided if the space-time code (STC) satisfies specific decoupling conditions; examples of such codes are orthogonal space-time block codes and quasi-orthogonal codes. In this letter, we study decoupling conditions for a linear full-diversity STC. Quasi-decoupled codes are proposed as a partially decoupled full-diversity STC of any rate for any number of transmit antennas with minimum decoding delay. By optimizing the coding gain of quasi-decoupled codes, it is shown that quasi-orthogonal codes have competitive performance with respect to the Alamouti code, and the more-recent threaded algebraic space-time (TAST) codes and ABBA codes. A general full-diversity decoupling condition is considered, and the general solution to this case, which also encompasses previously known orthogonal STCs, is derived

On Orthogonal Designs and Space-Time Codes

Orthogonal-design-based space-time (ODST) codes of size (n/spl times/n) offer maximum diversity gain advantage and a simple yet optimal decoding algorithm under an arbitrary signal alphabet or constellation A. However, these designs only exist for n=2, 4, 8 when A is real and for n=2 when A is complex. In this letter, we address the question of the existence of ODST codes of other sizes when A is restricted to be a proper subset of either real or complex numbers. We refer to these as restricted-alphabet ODST (RA-ODST) codes. We show that real RA-ODST codes of size greater than 8 that also guarantee maximum diversity advantage do not exist. Without the diversity requirement, RA-ODST codes exist only when A={a,-a}, 0<a/spl isin/R. Examples of such codes are provided. In the complex case, under the added requirement of maximum diversity advantage, we prove the nonexistence of complex RA-ODST codes under fairly simple assumptions regarding the signal alphabet.

Construction of optimal Type IV self-dual codes over F/sub 2/+uF/sub 2/

Previously, Type IV self dual codes over the ring F/sub 2/+uF/sub 2/ of order 4 have been introduced. In this paper, we construct five optimal Type IV self-dual codes which are the first examples of such codes for that length. We also give several new optimal type IV self-dual codes, and Type IV self-dual codes which improve the upper bounds for minimum weights.

Where Does it Hurt? Identifying the Real Concerns in the Ethics of Reference Service

What is the right thing to do-for our clients, for ouremployers,for our government and society, for, our colleagues and ourselves? In theory no subject would seem to be of more importance to reference librarians than the ethical considerations that underlie theiractivities. In fact the topic gets little attention.' Bernard Vavrek, who is himself one of the few people to make a serious study of ethical issues, has deplored reference librarians' lack of interest in this subject.* His concern is commendable but the sad reality is that, considering the way in which the subject has usually been treated and presented, the apathy with which it has been received is understandable and perhaps even warranted. The most common type of presentation has taken the form of a library association promulgating an official code or formal statement which presumes to set forth the principles and precepts by which the librarian is to be guided into righteous behavior. Some examples of such codes or statements are those issued by the American Library Association (1938 and 1975)? the Institute of Professional Librarians of Ontario (1975): the Ohio Library Association (1976),J the California Library Association (1976): the American Library Association (the 1979 revision of the 1975 version)? and the Library Association (1980).8 The failings of these documents are not difficult to detect. In a number of instances, notably that of the ALA Code of Ethics of 1938, the writing itself is so bad as to have prompted severe criticism. The vague idealism,'' fatuous adjurationsl0 and pompous platitudes11 of such a statement are reason enough for ignoring it. Another problem is that the documents often mix up major points of principle with minor matters of etiquette.'* Stillanotherdrawback, in thiscase more the fault

Stacking techniques for linear codes

A procedure for subspace stacking is proposed, and it is proved that the technique is equally successful for both linear codes and anticodes. It is demonstrated that families of errorcorrecting codes may be constructed using the proposed technique for stacking linear codes. Examples of such codes are given with rates better than the best known codes of identical Hamming distance and the same number of information digits.

New families of error-correcting codes generated by modification of other linear binary block codes

Two procedures for modifying linear block error-correction codes are proposed. The first is based on the deletion of x rows and y columns from the parity-check matrix of a given (n, k) code, in such a way that the minimum Hamming distance of the resulting (n−y, k+x−y) code remains equal to that of the original. It is shown that if x is unity, then y may be as low asthe minimum Hamming distance of the (n, n−k) dual of the original (n, k) code. This procedure of deleting rows and columns when applied to the known linear binary block code yields a family of codes, some of which have better rates than those of the best previously known codes of identical Hamming distance and the same number of paritycheck digits. 17 examples of such new codes are derived and included in the paper. It is also shown that apart from appropriate slight modification, the coding and decoding algorithms for this family of modified codes are similar to those of the original code. The second proposed procedure of code modification entails lengthening the original (n, k) linear block code by annexing k′ message digits. If the original code is capable of correcting t random errors or less, then the resulting modified (n+k′,k+k′) code has a rate higher than that of the original (n, k) code. Moreover, its error-correcting capability is such that it will correct t random errors or less if at least one of these occurs in the block of k message digits and s random errors or less, where 1≤s&lt;t, if none of the errors occur in any of the k message digits. Five examples of such codes are given.

Permutation codes for discrete channels (Corresp.)

A class of nonlinear codes for discrete channels is defined with the use of permutation groups. Several examples of such codes are given; their parameters are similar to those for Reed-Solomon codes in some cases. Decoding algorithms are discussed.