On the number of $ t $-Lee-error-correcting codes

Nadja Willenborg; Anna-Lena Horlemann; Violetta Weger

doi:10.3934/amc.2023055

Article Contents

2024, Volume 18, Issue 2: 567-594. Doi: 10.3934/amc.2023055

This issue Previous Article Sidon sets, sum-free sets and linear codes Next Article Criteria for the construction of MDS convolutional codes with good column distances

On the number of $ t $-Lee-error-correcting codes

1.
University of St.Gallen, Switzerland
2.
Technical University of Munich, Germany

^*Corresponding author: Nadja Willenborg
^*Corresponding author: Nadja Willenborg

Received: March 2023

Revised: November 2023

Early access: December 2023

Published: April 2024

The first author is supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 899987

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Abstract

We consider $ t $-Lee-error-correcting codes of length $ n $ over the residue ring $ \mathbb{Z}_{m} : = \mathbb{Z}/m\mathbb{Z} $ and determine upper and lower bounds on the number of $ t $-Lee-error-correcting codes. We use two different methods, namely estimating isolated nodes on bipartite graphs and the graph container method. The former gives density results for codes of fixed size and the latter for any size. This confirms some recent density results for linear Lee metric codes and provides new density results for nonlinear codes. To apply a variant of the graph container algorithm we also investigate some geometrical properties of the balls in the Lee metric.

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Mathematics Subject Classification: Primary: 94B25, 94B65; Secondary: 05C90.

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