Near MDS poset codes and distributions

We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.Comment: 13 pages, 1 figur

Bounds on the size of codes

In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way

Block Codes on Pomset Metric

Given a regular multiset $M$ on $[n]=\{1,2,\ldots,n\}$, a partial order $R$ on $M$, and a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i) = k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, we define a pomset block metric $d_{(Pm,\pi)}$ on the direct sum $\mathbb{Z}_{m}^{k_1} \oplus \mathbb{Z}_{m}^{k_2} \oplus \ldots \oplus \mathbb{Z}_{m}^{k_n}$ of $\mathbb{Z}_{m}^{N}$ based on the pomset $\mathbb{P}=(M,R)$. This pomset block metric $d_{(Pm,\pi)}$ extends the classical pomset metric which accommodate Lee metric introduced by I. G. Sudha and R. S. Selvaraj, in particular, and generalizes the poset block metric introduced by M. M. S. Alves et al, in general, over $\mathbb{Z}_m$. We find $I$-perfect pomset block codes for both ideals with partial and full counts. Further, we determine the complete weight distribution for $(P,\pi)$-space, thereby obtaining it for $(P,w)$-space, and pomset space, over $\mathbb{Z}_m$. For chain pomset, packing radius and Singleton type bound are established for block codes, and the relation of MDS codes with $I$-perfect codes is investigated. Moreover, we also determine the duality theorem of an MDS $(P,\pi)$-code when all the blocks have the same length.Comment: 17 Page

Pseudometrics, The Complex of Ultrametrics, and Iterated Cycle Structures

Every set X, finite of cardinality n say, carries a set M(X) of all possible pseudometrics. It is well known that M(X) forms a convex polyhedral cone whose faces correspond to triangle inequalities. Every point in a convex cone can be expressed as a conical sum of its extreme rays, hence the interest around discovering and classifying such rays. We shall give examples of extreme rays for M(X) exhibiting all integral edge lengths up to half the cardinality of X. By intersecting the cone with the unit cube we obtain the convex polytope of bounded-by-one pseudometrics BM(X). Analogous to extreme rays, every point in a convex polytope arises as a convex combination of extreme points. Extreme rays of BM(X) give rise to very special extreme points of ̄BM(X) as we may normalize a nonzero pseudometric to make its largest distance 1. We shall give a simple and complete characterization of extremeness for metrics with only edge lengths equal to 1/2 and 1. Then we shall use this characterization to give a decomposition result for the upper half of BM(X). BM(X) contains the set of bounded-by-1 pseudoultrametrics, U(X). Ultrametrics satisfy a stronger version of the triangle inequality, and have an interesting structure expressed in terms of partition chains. We will describe the topology of U(X) and its subset of scaled ultrametrics, SU(X), up to homotopy equivalence. Every permutation on a set X can be written as a product of disjoint cycles that cover X. In this way, a permutation generalizes a partition. An iterated cycle structure (ICS) will then be the associated generalization of a partition chain. Analogously, we will compute the “Euler-characteristic” of the set of iterated cycle structures

Ordered Covering Arrays and Upper Bounds on Covering Codes in NRT spaces

This work shows several direct and recursive constructions of ordered covering arrays using projection, fusion, column augmentation, derivation, concatenation and cartesian product. Upper bounds on covering codes in NRT spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.Comment: 27 page