Block $ LS $–poset codes over $  {\mathbb{Z}}_m^n $: Perfect codes, Singleton bound, and MDS characterization

Thitarie Rungratgasame; Phichet Jitjankarn; Thitarie Rungratgasame; Phichet Jitjankarn

doi:10.3934/math.2026605

AIMS Mathematics

2026, Volume 11, Issue 5: 14735-14756. doi: 10.3934/math.2026605

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Block $ LS $–poset codes over $ {\mathbb{Z}}_m^n $: Perfect codes, Singleton bound, and MDS characterization

Thitarie Rungratgasame ¹,
Phichet Jitjankarn ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Srinakharinwirot University, 114 Sukumvit 23 Wattana District, Bangkok 10110, Thailand
2.
Division of Mathematics, School of Science, Walailak University, 222 Thasala District, Nakhon Si Thammarat 80161, Thailand

Received: 11 March 2026 Revised: 20 April 2026 Accepted: 27 April 2026 Published: 25 May 2026
MSC : 05E99, 06A07, 20K30, 94B05, 94B65

We introduce a block $ LS $–poset (partially ordered set) metric on $ {\mathbb{Z}}_m^n $ constructed from a block decomposition of $ {\mathbb{Z}}_m^n $, a poset structure on the block indices, and the lattice of subgroups of $ {\mathbb{Z}}_m $ arising from the prime factorization of $ m $. Using a multiset representation associated with this subgroup lattice, we define the block $ LS $–poset weight and show that the induced distance is a metric on $ {\mathbb{Z}}_m^n $. We investigate the geometry of $ r $-balls and $ \mathcal I $-balls and establish their fundamental properties, including linearity, translation invariance, and duality. These structural results lead to characterizations of $ \mathcal I $-perfect block $ LS $–poset codes for ideals with full count and partial count. We further derive a Singleton-type bound for block $ LS $–poset codes and introduce the notions of maximum distance separable (MDS) and partial-MDS block $ LS $–poset codes. Connections among perfect codes, MDS codes, and $ r $-perfect codes are also examined for certain classes of posets.
- block $ LS $–poset metric,
- poset block code,
- perfect code,
- $ r $-perfect code,
- $ \mathcal I $-perfect code,
- MDS code,
- partial-MDS code,
- Singleton bound,
- subgroup lattice,
- multiset
Citation: Thitarie Rungratgasame, Phichet Jitjankarn. Block $ LS $–poset codes over $ {\mathbb{Z}}_m^n $: Perfect codes, Singleton bound, and MDS characterization[J]. AIMS Mathematics, 2026, 11(5): 14735-14756. doi: 10.3934/math.2026605

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Abstract

We introduce a block $ LS $–poset (partially ordered set) metric on $ {\mathbb{Z}}_m^n $ constructed from a block decomposition of $ {\mathbb{Z}}_m^n $, a poset structure on the block indices, and the lattice of subgroups of $ {\mathbb{Z}}_m $ arising from the prime factorization of $ m $. Using a multiset representation associated with this subgroup lattice, we define the block $ LS $–poset weight and show that the induced distance is a metric on $ {\mathbb{Z}}_m^n $. We investigate the geometry of $ r $-balls and $ \mathcal I $-balls and establish their fundamental properties, including linearity, translation invariance, and duality. These structural results lead to characterizations of $ \mathcal I $-perfect block $ LS $–poset codes for ideals with full count and partial count. We further derive a Singleton-type bound for block $ LS $–poset codes and introduce the notions of maximum distance separable (MDS) and partial-MDS block $ LS $–poset codes. Connections among perfect codes, MDS codes, and $ r $-perfect codes are also examined for certain classes of posets.

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