Weights which respect support and NN-decoding

Machado, Roberto Assis; Firer, Marcelo

doi:10.1109/isit.2019.8849342

Cited by 1 publication

(1 citation statement)

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“…The set T S(n) of all possible TS-metrics on F n 2 is not well described or studied, but it contains two well understood large families that can be used as bricks out of which every TS-metric can be build (for details see [29]). Those are the families of poset and combinatorial metrics, which we now introduce.…”

Section: A Ts-metricsmentioning

confidence: 99%

Obtaining Binary Perfect Codes Out of Tilings

Miyamoto

Firer

2020

IEEE Trans. Inform. Theory

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A tiling of the n-dimensional Hamming cube gives rise to a perfect code (according to a given metric) if the basic tile is a metric ball. We are concerned with metrics on the n-dimensional Hamming cube which are determined by a weight which respects support of vectors (TS-metrics). We consider the known tilings of the Hamming cube and first determine which of them give rise to a perfect code. In the sequence, for those tilings that satisfy this condition, we determine all the TS-metrics that turns it into a perfect code. We also propose the construction of new perfect codes obtained by the concatenation of two smaller ones. I. INTRODUCTIONPerfect codes have been extensively studied in the literature, due to the optimality condition imbued in its definition and to the interesting challenges they pose. Nevertheless, considering the Hamming metric, perfect codes are rare and they are classified in the case of linear binary codes: It has the same parameters of binary repetition codes with odd length, q-ary Hamming codes, binary or ternary Golay codes, [16], [21]. The situation is not the same for the Lee metric. There are few results (see [17], [18], [33]), but there is not a complete characterization. Two good surveys about perfect codes in Hamming metric are [22], [28].The most primary definition of a perfect code is the geometrical one: a code is perfect if its packing radius equals its covering radius. This definition is interesting in our setting since it depends directly on the metric invariants, which are naturally defined in general settings. To be more explicit, given a metric space (X, d) and a subset C ⊂ X, we define its packing radius R d,pack (C) as the maximal r such the balls of radius r centered at elements of C are disjoint and its covering radius R d,cov (C) as the minimum r such that the balls of radius r centered at elements of C covers the space X. The set C is called a d-perfect code if R d,pack (C) = R d,cov (C). Set this, it is understandable that the study of perfect codes can be undertaken considering a more vast family of metrics and it can be valuable to do so for any metric that has some relevance in the context of coding theory.Considering this scarcity of perfect codes under the Hamming metric, the introduction of the poset metrics by Brualdi et al. in 1995 [6] drawn the attention since, in general, there is a relative abundance of perfect codes (depending on the poset metric). The study of perfect codes in this context is done in different approaches. One of them is to fix a particular family of posets (chain, crown or hierarchical) and to classify all the perfect codes for the particular family, as done in [25], [26], [31].Another approach is to consider a family of well known codes and asks to classify all the poset metrics which turn the code to be perfect. This is what is done, for example, with the extended Hamming and Golay codes for poset metrics (in [23], [24]) and for poset-block metrics ( [2], [8]). Our approach resembles more the second one, but instead of looking at the ...

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Section: A Ts-metricsmentioning

confidence: 99%