Quasi-perfect Codes Research Articles

Perfect and Quasi-Perfect Codes for the Bosonic Classical-Quantum Channel

We explore perfect and quasi-perfect codes for the Bosonic channel, where information is generated by a laser and conveyed in the form of coherent states. In particular, we consider the phase-modulation codebook for coherent states in a Bosonic channel. We show that these phase-modulation codes are quasi-perfect as long as the cardinality of the code is the same as the dimension of the coherent states. These codes feature the smallest error probability among all codes of the same cardinality and the same dimension of the channel Hilbert space. We study the performance of these codes in terms of error probability, incorporating the degradation caused by a depolarizing or an erasure quantum channel.

Generalized Perfect Codes for Symmetric Classical-Quantum Channels

We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> -qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.

Some Binary Quasi-perfect Linear Codes Defined by APN Functions

In 2021, Tutdere proved that the covering radii R of a class of primitive binary cyclic codes with minimum distance strictly greater than an odd integer l satisfy r≤R≤l, where l, r are some integers depending on the given code. We here first discuss some equivalences of linear codes defined by Gold functions, which are quadratic APN (almost perfect nonlinear) functions. We then show that by applying the result of Tutdere one can find the covering radii of these quasi-perfect codes. In 2016, Li and Helleseth proved that the linear codes defined by the quadratic APN functions are quasi-perfect and they asked whether the linear codes defined by the non-quadratic APN functions are quasi-perfect or not. We here prove that the linear codes defined by some non-quadratic APN functions over the finite field〖 F〗_(2^m ) , for 1≤m≤8, are quasi-perfect, by computing the covering radii of these codes.

On the number of q-ary quasi-perfect codes with covering radius 2

In this paper we present a family of q-ary nonlinear quasi-perfect codes with covering radius 2. The codes have length $$n = q^m$$ and size $$ M = q^{n - m - 1}$$ where q is a prime power, $$q \ge 3$$ , m is an integer, $$m \ge 2$$ . We prove that there are more than $$q^{q^{cn}}$$ nonequivalent such codes of length n, for all sufficiently large n and a constant $$c = \frac{1}{q} - \varepsilon $$ .

The Error Probability of Generalized Perfect Codes via the Meta-Converse

We introduce a definition of perfect and quasi-perfect codes for discrete symmetric channels based on the packing and covering properties of generalized spheres whose shape is tilted using an auxiliary probability measure. This notion generalizes previous definitions of perfect and quasi-perfect codes and encompasses maximum distance separable codes. The error probability of these codes, whenever they exist, is shown to coincide with the estimate provided by the meta-converse lower bound. We illustrate how the proposed definition naturally extends to cover almost-lossless source-channel coding and lossy compression.

New Results on Binary Codes Obtained by Doubling Construction

Abstract Binary codes created by doubling construction, including quasi-perfect ones with distance d = 4, are investigated. All [17·2r−6, 17·2r−6 − r, 4] quasi-perfect codes are classified. Weight spectrum of the codes dual to quasi-perfect ones with d = 4 is obtained. The automorphism group Aut(C) of codes obtained by doubling construction is studied. A subgroup of Aut(C) is described and it is proved that the subgroup coincides with Aut(C) if the starting matrix of doubling construction has an odd number of columns. (It happens for all quasi-perfect codes with d = 4 except for Hamming one.) The properness and t-properness for error detection of codes obtained by doubling construction are considered.

Quasi-perfect codes in the $$\ell _p$$ ℓ p metric

We introduce the notion of degree of imperfection of a code in \(\mathbb {Z}^n\) with the \(\ell _p\) metric, to extend the so-called quasi-perfect codes. Through the establishment of bounds and computational approach, we determine all radii for which there are linear quasi-perfect codes for \(p = 2\) and \(n = 2, 3\). Numerical results concerning the codes with small degree of imperfection are also presented.

New Results on Codes Correcting Single Error of Limited Magnitude for Flash Memory

Some physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This motivated the application of the asymmetric limited magnitude error model in flash memory. In this paper, we present a new construction of quasi-perfect codes for such errors, and we also study the perfect codes with symmetric errors. Moreover, we show some nonexistence results on perfect codes for correcting single error of limited magnitude.

Quasi-perfect linear codes from planar and APN functions

Let p be a prime and m be a positive integer with m ? 3. Let f be a mapping from Fpm$\mathbb {F}_{p^{m}}$ to itself and Cf$\mathcal {C}_{f}$ be the linear code of length pm ? 1, whose parity-check matrix has its j-th column ?jf(?j)${\left [\begin {array}{c} \pi ^{j}\\ f(\pi ^{j}) \end {array} \right ]}$, where ? is a primitive element in Fpm$\mathbb {F}_{p^{m}}$ and j = 0, 1, ? , pm ? 2. In the case of p = 2, it is proved that Cf$\mathcal {C}_{f}$ has covering radius 3 when f(x) is a quadratic APN function. This gives a number of binary quasi-perfect codes with minimum distance 5. In the case that p is an odd prime, we show that for all known planar functions f(x), the covering radius of Cf$\mathcal {C}_{f}$ is equal to 2 if m is odd and 3 if m is even. Consequently, several classes of p-ary quasi-perfect codes are derived.

On the Equivalence Between Maximum Likelihood and Minimum Distance Decoding for Binary Contagion and Queue-Based Channels with Memory

We study the optimal maximum likelihood (ML) block decoding of general binary codes sent over two classes of binary additive noise channels with memory. Specifically, we consider the infinite and finite memory Polya contagion and queue-based channel models, which were recently shown to approximate well binary modulated correlated fading channels used with hard-decision demodulation. We establish conditions on the codes and channels parameters under which ML and minimum Hamming distance decoding are equivalent. We also present results on the optimality of classical perfect and quasi-perfect codes when used over the channels under ML decoding. Finally, we briefly apply these results to the dual problem of syndrome source coding with and without side information.

Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes have been considered for 2-D and 3-D Lee metric spaces. In this paper, constructive methods for obtaining quasi-perfect codes over metric spaces modeled by means of Gaussian and Eisenstein-Jacobi integers are given. The obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes. Moreover, they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels. Therefore, the results in this paper complement the constructions of perfect codes previously done for the same integer rings. Finally, decoding algorithms for the quasi-perfect codes obtained in this paper are provided and the relationship of the codes and the Lee metric ones is investigated.

A family of constacyclic ternary quasi-perfect codes with covering radius 3

In this paper a family of constacyclic ternary quasi-perfect linear block codes is presented. This family extends the result presented in a previous work by the first two authors, where the existence of codes with the presented parameters was stated as an open question. The codes have a minimum distance 5 and covering radius 3.

Classification of Niederreiter–Rosenbloom–Tsfasman Block Codes

Poset and block metrics were introduced in recent years as alternative metrics to study error correcting codes. Poset-block codes were introduced in 2008, encompassing both poset and block metrics. In this paper, we study a family of such metrics, the Niederreiter-Rosenbloom-Tsfasman block metrics. In this context, we classify the classes of equivalent codes, describe canonical representatives of each class and develop much of the classical theory of error correcting codes for Niederreiter-Rosenbloom-Tsfasman block codes, including determination of packing radius and classification of MDS and perfect codes, determination of covering radius and characterization of quasi-perfect codes, and the description of an algorithm for syndrome decoding.

On decoding binary perfect and quasi-perfect codes over markov noise channels

We study the decoding problem when a binary linear perfect or quasi-perfect code is transmitted over a binary channel with additive Markov noise. After examining the properties of the channel block transition distribution, we derive sufficient conditions under which strict maximum-likelihood decoding is equivalent to strict minimum Hamming distance decoding when the code is perfect. Additionally, we show a near equivalence relationship between strict maximum likelihood and strict minimum distance decoding for quasi-perfect codes for a range of channel parameters and the code's minimum distance. As a result, an improved (complete) minimum distance decoder is proposed and simulations illustrating its benefits are provided.

Generating abelian groups by addition only

We define the positive diameter of a finite group G with respect to a generating set A ⊆ G to be the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A . This invariant, which we denote by , can be interpreted as the diameter of the Cayley digraph induced by A on G . In this paper we study the positive diameters of a finite abelian group G with respect to its various generating sets A . More specifically, we determine the maximum possible value of and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of A subject to the condition that is “not too small”. Conceptually, the problems studied are closely related to our earlier work [Klopsch and Lev, J. Algebra 261: 145–171, 2003] and the results obtained shed a new light on the subject. Our original motivation came from connections with caps, sum-free sets, and quasi-perfect codes.

Binary and Ternary Linear Quasi-Perfect Codes With Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.

Quasi-Perfect Linear Codes With Minimum Distance $4$

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n,n-m,4]q-code when either 1) q=16, m ges 5, m odd, or 2) q=2i, 7 les i les 15, m ges 4, or 3) q=22lscr , lscr ges 8, m ges 5, m odd. As quasi-perfect [n,n-m,4]q-codes and complete n-caps in projective spaces PG(m-1,q) are equivalent objects, new upper bounds on the size of the smallest complete cap in PG(m-1,q) are obtained

Quasi-Perfect Codes With Small Distance

The main purpose of this paper is to give bounds on the length of the shortest and longest binary quasi-perfect codes with a given Hamming distance, covering radius, and redundancy. We consider codes with Hamming distance 4 and 5 and covering radius 2 and 3, respectively. We discuss the blockwise direct sum (BDS) construction which has an important role in finding these bounds.

A Tight Upper Bound on the BER of Linear Systematic Block Codes

We present in this letter new upper bounds for linear and systematic block codes. Both perfect and quasi-perfect codes are considered. Numerical results show that our new bounds give much tighter results compared to the classical union-Chernoff bound. In addition, these bounds are more accurate than the leading term approximations especially at low signal-to-noise ratio.

How long does it take to generate a group?

The diameter of a finite group G with respect to a generating set A is the smallest non-negative integer n such that every element of G can be written as a product of at most n elements of A ∪ A −1 . We denote this invariant by diam A ( G ). It can be interpreted as the diameter of the Cayley graph induced by A on G and arises, for instance, in the context of efficient communication networks. In this paper we study the diameters of a finite Abelian group G with respect to its various generating sets A . We determine the maximum possible value of diam A ( G ) and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of A subject to the condition that diam A ( G ) is “not too small”. Connections with caps, sum-free sets, and quasi-perfect codes are discussed.