Poset Metrics Research Articles

MacWilliams Extension Property With Respect to Weighted Poset Metric

MacWilliams Extension Property With Respect to Weighted Poset Metric

Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

Fourier-Reflexive Partitions Induced by Poset Metric

Let H be the cartesian product of a family of finite abelian groups indexed by a finite set Ω. A given poset (i.e., partially ordered set) P = (Ω, ≼P) gives rise to a poset metric on H, which further leads to a partition Q(H, P) of H. We prove that if Q(H, P) is Fourier-reflexive, then its dual partition Λ coincides with the partition of Ĥ induced by P, the dual poset of P, and moreover, P is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in [5]. We also show that with some other assumptions, Λ is finer than the partition of Ĥ induced by P. In addition, we give some necessary and sufficient conditions for P to be hierarchical, and for the case that P is hierarchical, we give an explicit criterion for determining whether two codewords in Ĥ belong to the same block of Λ. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

MDS and I-Perfect Codes in Pomset Metric

Recently pomset metric was introduced to accommodate Lee metric, thereby generalizing poset metrics for codes over $\mathbb {Z}_{{m}}$ . This work mainly studies results on Maximum distance separable (MDS) pomset codes and perfect codes over $\mathbb {Z}_{{m}}$ . If I is an ideal in a poset, the concept of I -perfect codes that was introduced to study MDS poset codes is now extended to the study of MDS pomset codes as well. But, unlike the ideals in posets, here the ideals in pomsets are considered as those with full count and those with partial count. In fact, the I -balls are no more linear when the ideal I is with partial count. This makes the investigation of the relationship of I -perfect codes with MDS pomset codes little different from that with MDS poset codes. We thoroughly study the I -balls where I is with partial count. Singleton type bound is established for codes with pomset metric and the connection of MDS codes with I -perfect codes is investigated upon. Then, we prove the duality theorem for codes with chain pomset. Finally, we determine the weight distribution of MDS pomset codes when the pomset is a chain.

General Approach to Poset and Additive Metrics

Let $P=([n],\leq _{P})$ be a poset on $[n] = \{1,2,\ldots,n\}$ , $\mathbb {F}_{q}^{n}$ be the linear space of $n$ -tuples over a finite field $\mathbb {F}_{q}$ and $w$ be a weight on $\mathbb {F}_{q}$ . In this paper we consider metrics on $\mathbb {F}_{q}^{n}$ which are induced by posets over $[n]$ and weights over $\mathbb {F}_{q}$ . Such family of metrics extend both the additive metric induced by the weight $w$ (when the poset is an anti-chain) and the poset metrics (when the weight is the Hamming weight). Furthermore, the pomset metrics is also a particular case of our construction, consequently, we provide a simpler approach to these metrics without using the multiset structure originally proposed. For the general case, we provide a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we (re)obtain the groups of linear isometries of the poset, pomset and additive metric spaces. When considering a chain order, we develop, for codes on these spaces, several of the invariants and properties found in the classical coding theory. Our construction of metrics based on partial orders and any weight over the base field, highlights the dependence of the poset metric over $\mathbb {F}_{q}^{n}$ with the Hamming metric on $\mathbb {F}_{q}$ and the additive property of its extension on $\mathbb {F}_{q}^{n}$ .

Weighted Posets and Digraphs Admitting the Extended Hamming Code to be a Perfect Code

Recently, Etzion et al. introduced metrics on ${\mathbb{F}}_{2}^{n}$ based on directed graphs on $n$ vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code $\tilde {\mathcal {H }}_{3}$ to be a two-perfect code. We also construct some families of such structures for any $k \geq 3$ , which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps.

Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces

Alves, Panek and Firer (Error-block codes and poset metrics, Adv. Math. Commun., 2 (2008), 95-111) classified all poset block structures which turn the [8,4,4] extended binary Hamming code into a 1-perfect poset block code. However, the proof needs corrections that are supplied in this paper. We provide a counterexample to show that the extended binary Golay code is not 1-perfect for the proposed poset block structures. All poset block structures turning the extended binary and ternary Golay codes into 1-perfect codes are classified.

Characterization of Metrics Induced by Hierarchical Posets

In this paper, we consider metrics determined by hierarchical posets and give explicit formulas for the main parameters of a linear code: the minimum distance and the packing, covering, and Chebyshev radii of a code. We also present ten characterizations of hierarchical poset metrics, including new characterizations and simple new proofs to the known ones.

Codes with a pomset metric and constructions

Brualdi’s introduction to the concept of poset metric on codes over $$\mathbb {F}_{q}$$ paved a way for studying various metrics on $$\mathbb {F}_{q}^{n}$$ . As the support of vector x in $$\mathbb {F}_{q}^{n}$$ is a set and hence induces order ideals and metrics on $$\mathbb {F}_{q}^{n}$$ , the poset metric codes could not accommodate Lee metric structure due to the fact that the support of a vector with respect to Lee weight is not a set but rather a multiset. This leads the authors to generalize the poset metric structure on to a pomset (partially ordered multiset) metric structure. This paper introduces pomset metric and initializes the study of codes equipped with pomset metric. The concept of order ideals is enhanced and pomset metric is defined. Construction of pomset codes are obtained and their metric properties like minimum distance and covering radius are determined.

Perfect codes in poset block spaces

There is a limited class of perfect codes with respect to the classical Hamming metric. There are other kind of metrics with respect to which perfect codes have been investigated viz. poset metric, block metric and poset block metric. Given the minimal elements of a poset, a necessary and sufficient condition for [Formula: see text]-perfectness of a poset block code has been derived. A necessary and sufficient condition for a poset block code to be [Formula: see text]-perfect has also been considered. Further, for each [Formula: see text], [Formula: see text], a sufficient condition that ensures the existence of a poset block structure which turns a given code into an [Formula: see text]-perfect poset block code has been obtained. Several illustrations of well known codes to be [Formula: see text]-perfect for specific values of [Formula: see text] have been explored.

The packing radius of a poset block code

We investigate the properties of the packing radius of a code with respect to poset block metric. In the process, we have addressed a few minor errors in the paper, “The packing radius of a code and partitioning problems: The case for poset metrics”, in Proc. IEEE Int. Symp. Information Theory (2014), pp. 2954–2958 by D’Oliveira and Firer.

The packing radius of a code and partitioning problems: The case for poset metrics on finite vector spaces

Until this work, the packing radius of a poset code was only known in the cases where the poset was a chain, a hierarchy, a union of disjoint chains of the same size, and for some families of codes. Our objective is to approach the general case of any poset. To do this, we will divide the problem into two parts.The first part consists in finding the packing radius of a single vector. We will show that this is equivalent to a generalization of a well known NP-hard problem known as “the partition problem”. Then, we will review the main results known about this problem giving special attention to the algorithms to solve it. The main ingredient to these algorithms is what is known as the differentiating method, which we will extend to the general case.The second part consists in finding the vector that determines the packing radius of the code. For this, we will show how it is sometimes possible to compare the packing radius of two vectors without calculating them explicitly.

POSETS ADMITTING THE LINEARITY OF ISOMETRIES

Abstract. In this paper, we deal with a characterization of the posetswith the property that every poset isometry of F nq fixing the origin is alinear map. We say such a poset to be admitting the linearity of isome-tries. We show that a poset P admits the linearity of isometries over F nq if and only if P is a disjoint sum of chains of cardinality 2 or 1 whenq= 2, or P is an anti-chain otherwise. 1. IntroductionLet F q be a finite field with q elements, and F nq the vector space of n-tuplesover F q . In 1995, Brualdi et al. [1] introduced a non-Hamming metric on F n which is associated to an arbitrary poset on [n] = {1,2,...,n}. It is calleda poset metric. The poset metric spaces have been extensively studied in[1, 3, 4, 5, 8]. We briefly introduce basic notions for poset metric on F nq .Let P = ([n],≤) be a poset on [n] of coordinate positions of vectors on F nq .A subset I of P is called an order ideal (or a down-set) if x ∈ I and y ≤ ximply y ∈ I. For an arbitrary subset A of P, we denote by hAi the smallestorder ideal of P containing A. The P-weight of a vector u = (u

POSET METRICS ADMITTING ASSOCIATION SCHEMES AND A NEW PROOF OF MACWILLIAMS IDENTITY

It is known that being hierarchical is a necessary and suffi- cient condition for a poset to admit MacWilliams identity. In this paper, we completely characterize the structures of posets which have an associ- ation scheme structure whose relations are indexed by the poset distance between the points in the space. We also derive an explicit formula for the eigenmatrices of association schemes induced by such posets. By using the result of Delsarte which generalizes the MacWilliams identity for linear codes, we give a new proof of the MacWilliams identity for hierarchical linear poset codes.

Canonical- systematic form for codes in hierarchical poset metrics

In this work we present a canonical-systematic form of a generator matrix for linear codes whith respect to a hierarchical poset metric on the linear space $\mathbb F_q^n$. We show that up to a linear isometry any such code is equivalent to the direct sum of codes with smaller dimensions. The canonical-systematic form enables to exhibit simple expressions for the generalized minimal weights (in the sense defined by Wei), the packing radius of the code, characterization of perfect codes and also syndrome decoding algorithm that has (in general) exponential gain when compared to usual syndrome decoding.

Duality for Poset Codes

In this paper, we extend Wei's Duality Theorem, relating the generalized Hamming weight hierarchy of a code to the hierarchy of the dual code, to the scope of codes with poset metrics. As a consequence of this duality theorem, we prove some results concerning discrepancy of codes and chain condition for generalized weights.

The Poset Metrics That Allow Binary Codes of Codimension $m$ to be $m$-, $(m-1)$-, or $(m-2)$-Perfect

A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of R-perfect poset codes for some R in the case of the crown poset and in the case of the union of disjoin chains. Index terms: perfect codes, poset codes

Projective systems and perfect codes with a poset metric

The notion of a projective system, defined as a set X of n-points in a projective space over a finite field, was introduced by Tsfasman and Vlǎdut. By this notion, the weight distribution of a nondegenerate linear code can be computed by the configuration of X and hyperplanes in the dual projective space of C. In this paper, we construct a projective system X defined as a set of n-linear subspaces on the dual projective space of a poset code. This gives a natural one-to-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of X and hyperplanes in the dual projective space of C. Finally, we provide an algorithm for finding perfect poset codes.

Codes with a poset metric

Niederreiter generalized the following classical problem of coding theory: given a finite field F q and integers n > k ⩾ 1, find the largest minimum distance achievable by a linear code over F q of length n and dimension k. In this paper we place this problem in the more general setting of a partially ordered set and define what we call poset-codes. In this context, Niederreiter's setting may be viewed as the disjoint union of chains. We extend some of Niederreiter's bounds and also obtain bounds for posets which are the product of two chains.