Construction Of Error-correcting Codes Research Articles

Independent Neighbourhoods of Sets in Bn Groups

The paper considers the structural problems and cardinality problems of pairs of non-empty subsets with certain restrictions with respect to the space where the sum is by modulo 2. The described subsets are related, in many cases, to the construction of error-correcting codes on additive communication channels, and to the quantitative bounds of their cardinality.

Solving Xq+1 + X + a = 0 over finite fields

Solving the equation Pa(X):=Xq+1+X+a=0 over the finite field FQ, where Q=pn,q=pk and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [2], the construction of difference sets with Singer parameters [8], determining cross-correlation between m-sequences [9,15] and the construction of error-correcting codes [5], as well as speeding up the index calculus method for computing discrete logarithms on finite fields [11,12] and on algebraic curves [18].Subsequently, in [3,13,14,6,4,16,7,19], the FQ-zeros of Pa(X) have been studied. It was shown in [3] that their number is 0, 1, 2 or pgcd⁡(n,k)+1. Some criteria for the number of the FQ-zeros of Pa(x) were found in [13,14,6,16,19]. However, while the ultimate goal is to identify all the FQ-zeros, even in the case p=2, it was solved only under the condition gcd⁡(n,k)=1[16].We discuss this equation without any restriction on p and gcd⁡(n,k). Criteria for the number of the FQ-zeros of Pa(x) are proved by a new methodology. For the cases of one or two FQ-zeros, we provide explicit expressions for these rational zeros in terms of a. For the case of pgcd⁡(n,k)+1 rational zeros, we provide a parametrization of such a's and express the pgcd⁡(n,k)+1 rational zeros by using that parametrization.

Efficient and Explicit Balanced Primer Codes

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing.

Error-correcting codes from k-resolving sets

We demonstrate a construction of error-correcting codes from graphs by means of $k$-resolving sets, and present a decoding algorithm which makes use of covering designs. Along the way, we determine the $k$-metric dimension of grid graphs (i.e. Cartesian products of paths).

Codes in the Space of Multisets—Coding for Permutation Channels With Impairments

Motivated by communication channels in which the transmitted sequences are subject to random permutations, as well as by certain DNA storage systems, we study the error control problem in settings where the information is stored/transmitted in the form of multisets of symbols from a given finite alphabet. A general channel model is assumed in which the transmitted multisets are potentially impaired by insertions, deletions, substitutions, and erasures of symbols. Several constructions of error-correcting codes for this channel are described, and bounds on the size of optimal codes correcting any given number of errors derived. The construction based on the notion of Sidon sets in finite Abelian groups is shown to be optimal, in the sense of the asymptotic scaling of code redundancy, for any "error radius" and any alphabet size. It is also shown to be optimal in the stronger sense of maximal code cardinality in various cases.

Extending 3-bit Burst Error-Correction Codes With Quadruple Adjacent Error Correction

The use of error-correction codes (ECCs) with advanced correction capability is a common system-level strategy to harden the memory against multiple bit upsets (MBUs). Therefore, the construction of ECCs with advanced error correction and low redundancy has become an important problem, especially for adjacent ECCs. Existing codes for mitigating MBUs mainly focus on the correction of up to 3-bit burst errors. As the technology scales and cell interval distance decrease, the number of affected bits can easily extend to more than 3 bit. The previous methods are therefore not enough to satisfy the reliability requirement of the applications in harsh environments. In this paper, a technique to extend 3-bit burst error-correction (BEC) codes with quadruple adjacent error correction (QAEC) is presented. First, the design rules are specified and then a searching algorithm is developed to find the codes that comply with those rules. The ${H}$ matrices of the 3-bit BEC with QAEC obtained are presented. They do not require additional parity check bits compared with a 3-bit BEC code. By applying the new algorithm to previous 3-bit BEC codes, the performance of 3-bit BEC is also remarkably improved. The encoding and decoding procedure of the proposed codes is illustrated with an example. Then, the encoders and decoders are implemented using a 65-nm library and the results show that our codes have moderate total area and delay overhead to achieve the correction ability extension.

Combinatorial PCPs with Short Proofs

The Probabilistically Checkable Proof (PCP) theorem (Arora and Safra in J ACM 45(1):70---122, 1998; Arora et al. in J ACM 45(3):501---555, 1998) asserts the existence of proofs that can be verified by reading a very small part of the proof. Since the discovery of the theorem, there has been a considerable work on improving the theorem in terms of the length of the proofs, culminating in the construction of PCPs of quasi-linear length, by Ben-Sasson and Sudan (SICOMP 38(2):551---607, 2008) and Dinur (J ACM 54(3):241---250, 2007). One common theme in the aforementioned PCP constructions is that they all rely heavily on sophisticated algebraic machinery. The aforementioned work of Dinur (2007) suggested an alternative approach for constructing PCPs, which gives a simpler and arguably more intuitive proof of the PCP theorem using combinatorial techniques. However, this combinatorial construction only yields PCPs of polynomial length and is therefore inferior to the algebraic constructions in this respect. This gives rise to the natural question of whether the proof length of the algebraic constructions can be matched using the combinatorial approach. In this work, we provide a combinatorial construction of PCPs of length $${n\cdot\left(\log n\right)^{O(\log\log n)}}$$n·lognO(loglogn), coming very close to the state-of-the-art algebraic constructions (whose proof length is $${n\cdot\left(\log n\right)^{O(1)}}$$n·lognO(1)). To this end, we develop a few generic PCP techniques which may be of independent interest. It should be mentioned that our construction does use low-degree polynomials at one point. However, our use of polynomials is confined to the construction of error-correcting codes with a certain simple multiplication property, and it is conceivable that such codes could be constructed without the use of polynomials. In addition, we provide a variant of the main construction that does not use polynomials at all and has proof length $${n^{4} \cdot\left(\log n\right)^{O(\log\log n)}}$$n4·lognO(loglogn). This is already an improvement over the aforementioned combinatorial construction of Dinur.

REGULAR METHOD FOR SYNTHESIS OF BASIC BENT-SQUARES OF RANDOM ORDER

Рассматриваются вопросы конструирования классов максимально нелинейных булевых бент-функций произвольной длины N = 2k (k = 2, 4, 6, …) на основе их спектрального представления – бент-квадратов Агиевича. Данные совершенные алгебраические конструкции являются основой для построения многих криптографических примитивов, таких как генераторы псевдослучайных ключевых последовательностей, криптографические S-блоки подстановки и т. д. Бент-функции находят свое применение для построения C-кодов в системах с кодовым разделением каналов, которые обладают минимально возможным значением пик-фактора k = 1, а также для построения систем ортогональных бифазных сигналов и помехоустойчивых кодов. Все многочисленные применения бент-функций связаны с теорией их синтеза. Однако регулярные методы синтеза полных классов бент-функций произвольной длины N = 2k в настоящее время неизвестны. В статье предложен регулярный метод синтеза базовых бент-квадратов Агиевича произвольного порядка n на основе регулярного оператора диадного сдвига. Выполнена классификация полного множества спектральных векторов длин (l = 8, 16, …) на основе критерия максимального абсолютного значения и набора абсолютных значений спектральных компонент. Показано, что любой спектральный вектор может быть основой для построения бент-квадрата. Обобщены результаты синтеза бент-квадратов Агиевича порядка n = 8, показано, что существуют только три базовых бент-квадрата для данного порядка, тогда как еще пять могут быть получены с помощью операции ступенчато циклического сдвига. Синтезированы все базовые бент-квадраты порядкаn = 16, позволяющие построение бент-функций длиной N = 256. Полученные базовые бент-квадраты могут служить как для непосредственного синтеза бент-функций и их практического использования, так и для проведения дальнейших исследований с целью синтеза новых структур бент-квадратов для порядков n = 16, 32, 64, …

Almost Euclidean subspaces of ℓ 1 N VIA expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ ℝN of dimension (1 - o(1))N such that for every x ∈ X, {display equation}. If we are allowed to use N1/log log N ≤ N°(1) random bits and dim(X) ≥ (1 - η)N for any fixed constant η, the lower bound can be further improved to (log N)-°(1) √N||x||2. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

Minimum distance of relative Reed–Muller codes

We describe a construction of error-correcting codes on a fibration over a curve defined over a finite field, which may be considered as a relative version of the classical Reed–Muller code. In the case of complete intersections in a projective bundle, we give an explicit lower bound for the minimum distance.

Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy

In this paper, we present error-correcting codes that achieve the information-theoretically best possible tradeoff between the rate and error-correction radius. Specifically, for every 0 < R < 1 and epsiv < 0, we present an explicit construction of error-correcting codes of rate that can be list decoded in polynomial time up to a fraction (1- R - epsiv) of worst-case errors. At least theoretically, this meets one of the central challenges in algorithmic coding theory. Our codes are simple to describe: they are folded Reed-Solomon codes, which are in fact exactly Reed-Solomon (RS) codes, but viewed as a code over a larger alphabet by careful bundling of codeword symbols. Given the ubiquity of RS codes, this is an appealing feature of our result, and in fact our methods directly yield better decoding algorithms for RS codes when errors occur in phased bursts. The alphabet size of these folded RS codes is polynomial in the block length. We are able to reduce this to a constant (depending on epsiv) using existing ideas concerning ldquolist recoveryrdquo and expander-based codes. Concatenating the folded RS codes with suitable inner codes, we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard GMD decoding.

Error-correcting codes on projective bundles

Let C be a nonsingular projective curve defined over a finite field. We give a construction of error-correcting codes on the projective bundle P ( E ) → C associated to a vector bundle E on C.

Codes on Grassmann bundles and related varieties

We give a construction of error-correcting codes from Grassmann bundles associated to a vector bundle on a curve defined over a finite field F q . We also consider codes from fibered varieties whose fibers are quadric or Hermitian varieties.

Guest column

The central algorithmic problem in coding theory is the construction of explicit error-correcting codes with good parameters together with fast algorithms for encoding a message and decoding a noisy received word into the correct message. Over the last decade, significant new developments have taken place on this problem using graph-based/combinatorial constructions that exploit the power of &lt;i&gt;expander graphs&lt;/i&gt;. The role of expander graphs in theoretical computer science is by now certainly well-appreciated. Here, we will survey some of the highlights of the use of expanders in constructions of error-correcting codes.

On the existence and construction of good codes with low peak-to-average power ratios

The first lower bound on the peak-to-average power ratio (PAPR) of a constant energy code of a given length n, minimum Euclidean distance and rate is established. Conversely, using a nonconstructive Varshamov-Gilbert style argument yields a lower bound on the achievable rate of a code of a given length, minimum Euclidean distance and maximum PAPR. The derivation of these bounds relies on a geometrical analysis of the PAPR of such a code. Further analysis shows that there exist asymptotically good codes whose PAPR is at most 8 log n. These bounds motivate the explicit construction of error-correcting codes with low PAPR. Bounds for exponential sums over Galois fields and rings are applied to obtain an upper bound of order (log n)/sup 2/ on the PAPRs of a constructive class of codes, the trace codes. This class includes the binary simplex code, duals of binary, primitive Bose-Chaudhuri-Hocquenghem (BCH) codes and a variety of their nonbinary analogs. Some open problems are identified.

Increasing the reliability of microprocessor-based measurement instruments

We consider construction of error-correcting codes to increase the probability of correction of single errors and detection of a maximum number of uncorrectable errors.

Fault-tolerant cube graphs and coding theory

Hypercubes, meshes, tori, and Omega networks are well-known interconnection networks for parallel computers. The structure of those graphs can be described in a more general framework called cube graphs. The idea is to assume that every node in a graph with q/sup l/ nodes is represented by a unique string of l symbols over GF(q). The edges are specified by a set of offsets, those are vectors of length l over GF(q), where the two endpoints of an edge are an offset apart. We study techniques for tolerating edge faults in cube graphs that are based on adding redundant edges. The redundant graph has the property that the structure of the original graph can be maintained in the presence of edge faults. Our main contribution is a technique for adding the redundant edges that utilizes constructions of error-correcting codes and generalizes existing ad hoc techniques.

Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing

A new general purpose algorithm for the solution of combinatorial optimization problems is presented. The new threshold accepting method is even simpler structured than the wellknown simulated annealing approach. The power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error-correcting codes. Moreover, deterministic (!) versions of the new heuristic turn out to perform nearly equally well, consuming only a fraction of the computing time of the stochastic versions. As an example, the deterministic threshold accepting method yields very-near-to-optimum tours for the famous 442-cities traveling salesman problem of Grötschel within 1 to 2 s of CPU time.

Error-correcting codes for byte-organized memory systems

Techniques are presented for the construction of error-correcting codes for semiconductor memory subsystems that are organized in a multibit-per-chip manner. These codes are capable of correcting all single-byte errors and detecting all double-byte errors, where a byte represents the number of bits that are fed from the same chip to the same codeword.