We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all codes are good coverings, then at most a vanishing fraction of codes are good packings. We also show that any specific maximal binary code is either a good packing or a good covering, in a certain well-defined sense.
A $λ$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $λ$ times. The well-known $r$-error-correcting codes correspond to the case $λ=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an…
We define generalized Hamming weights for almost affine codes. We show how various aspects and applications of generalized Hamming weights for linear codes, such as Wei duality, generalized Kung's bound, profiles, connection to wire-tap channels of type II, apply to the larger class of almost affine codes in general. In addition we discuss duality of almost affine codes,and of the smaller class of multilinear codes. We also give results about weight distributions of infinite series of almost affine codes, each series obtained from a fixed code by extending the code alphabet.
A randomized covering-packing duality between source and channel coding will be discussed by considering the source coding problem of coding a source with a certain distortion level and by considering a channel which communicates the source within a certain distortion level. An operational view of source-channel separation for communication with a fidelity criterion will be discussed in brief.
We consider linear codes over some fixed finite field extension over an arbitrary finite field. Gabidulin introduced rank metric codes, by endowing linear codes over the extension field with a rank weight over the base field and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of wiretap II codes in terms of generalized Hamming weights by Wei, Kurihara et al. defined some generalized rank weights and showed their relevance for secure network coding. In this paper, we derive a statement for generalized rank weights …
We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular $B_t$-sequences, $2$-fold Sidon sets, and sets avoiding arithmetic progression.
We consider Slepian-Wolf (SW) coding of multiple sources and extend the packing bound and the notion of perfect code from conventional channel coding to SW coding with more than two sources. We then introduce Hamming Codes for Multiple Sources (HCMSs) as a potential solution of perfect SW coding for arbitrary number of terminals. Moreover, we study the case with three sources in detail. We present the necessary conditions of a perfect SW code and show that there exists infinite number of HCMSs. Moreover, we show that for a perfect SW code with sufficiently long code length, the compression rat…
Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most $L$. For odd $L\ge 3$ an asymptotic upper bound on the rate of any such packing is proven. Resulting bound improves the best known bound (due to Blinovsky'1986) for rates below a certain threshold. Method is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn (that was used previously to improve the estimates of Blinovsky for $L=2$) and a Ramsey-theoretic technique of Blinovsky. As an applica…
Most design approaches for trellis-coded quantization take advantage of the duality of trellis-coded quantization with trellis-coded modulation, and use the same empirically-found convolutional codes to label the trellis branches. This letter presents an alternative approach that instead takes advantage of maximum-Hamming-distance convolutional codes. The proposed source codes are shown to be competitive with the best in the literature for the same computational complexity.
In $1991$, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei's duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei's duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galoi…
In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Recently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-…
This paper has two goals. The first one is to discuss good codes for packing problems in the Lee and Manhattan metrics. The second one is to consider weighing matrices for some of these coding problems. Weighing matrices were considered as building blocks for codes in the Hamming metric in various constructions. In this paper we will consider mainly two types of weighing matrices, namely conference matrices and Hadamard matrices, to construct codes in the Lee (and Manhattan) metric. We will show that these matrices have some desirable properties when considered as generator matrices for codes …
On the Generalized Packing and Covering Radii of Codes
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；Department of Electrical and Computer Engineering, McMaster University（麦克马斯特大学电气与计算机工程系）
We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.
This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their parameters, and investigate their asymptotic covering properties. Finally, we establish several identities that relate the rank weight distribution of a linear code to that of its dual code. One of our identities is the counterpart of the MacWilliams identity for the Hamming metric, and it has a different form from the identity by Delsarte.
Let $p$ be a prime number and $s> 0$ an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian $p$-extension of $\mathbb{F}_{p^{s}}(x)$. We determine their dimension and the exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list the exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for the self-duality and the quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, a…
First, we state a generalization of the minimum-distance bound for PIR codes. Then we describe a construction for linear PIR codes using packing designs and use it to construct some new 5-PIR codes. Finally, we show that no encoder (linear or nonlinear) for the binary $r$-th order Hamming code produces a 3-PIR code except when $r=2$. We use these results to determine the smallest length of a binary (possibly nonlinear) 3-PIR code of combinatorial dimension up to~6. A binary 3-PIR code of length 11 and size $2^7$ is necessarily nonlinear, and we pose the existence of such a code as an open prob…
Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every $n$ and $1\le M\le2^{n}$, determine the minimum average Hamming distance of binary codes with length $n$ and size $M$. Fu, Wei, and Yeung (2001) used linear programming duality to derive a lower bound on the minimum average distance. However, their linear programming approach was not completely exploited. In this paper, we improve Fu-Wei-Yeung's bound by finding a better feasible solution to their dual program. For fixed $0<a\le1/2$ and for $M=\left\lceil a2^{n}\right\rceil $, our feasible solution…
Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weigh…
We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $α(G) \leq \vartheta(G)$ between the independence number and the Lovász number of $G$ can be made arbitrarily tight: Precisely, the inequality \[ α(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \] becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales s…
We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play t…
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributio…
In the present paper we introduce and study finite point subsets of a special kind, called optimum distributions, in the n-dimensional unit cube. Such distributions are closely related with known (delta,s,n)-nets of low discrepancy. It turns out that optimum distributions have a rich combinatorial structure. Namely, we show that optimum distributions can be characterized completely as maximum distance separable codes with respect to a non-Hamming metric. Weight spectra of such codes can be evaluated precisely. We also consider linear codes and distributions and study their general properties i…
In this text we develop some aspects of Harder-Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder-Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids. We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistabili…
The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (\text…
Universal security over a network with linear network coding has been intensively studied. However, previous linear codes used for this purpose were linear over a larger field than that used on the network. In this work, we introduce new parameters (relative dimension/rank support profile and relative generalized matrix weights) for linear codes that are linear over the field used in the network, measuring the universal security performance of these codes. The proposed new parameters enable us to use optimally universal secure linear codes on noiseless networks for all possible parameters, as …
