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On Naisargik Images of Varshamov-Tenengolts and Helberg Codes

Kalp Pandya, Devdeep Shetranjiwala, Naisargi Savaliya, Manish K. Gupta

TL;DR

The article investigates how Naisargik maps from ${\mathbb{Z}}_4$ to ${\mathbb{Z}}_2^2$ affect binary and quaternary VT and Helberg codes, with eight maps used for VT and a single map for Helberg. It shows that VT images generally fail to preserve unit deletion-correcting properties, while Helberg codes gain increased deletion-correcting power under these maps (and their inverses). The work provides conjectures, partial proofs, and extensive data suggesting structured relations between codewords across alphabets, including a proposed one-to-one correspondence between certain quaternary and binary Helberg code families. These findings have implications for designing robust deletion/insertion-correcting codes for DNA storage and related channels, and they open paths for extending mappings to broader $q$-ary settings.

Abstract

The VT and Helberg codes, both in binary and non-binary forms, stand as elegant solutions for rectifying insertion and deletion errors. In this paper we consider the quaternary versions of these codes. It is well known that many optimal binary non-linear codes like Kerdock and Prepreta can be depicted as Gray images (isometry) of codes defined over $\mathbb{Z}_4$. Thus a natural question arises: Can we find similar maps between quaternary and binary spaces which gives interesting properties when applied to the VT and Helberg codes. We found several such maps called Naisargik (natural) maps and we study the images of quaternary VT and Helberg codes under these maps. Naisargik and inverse Naisargik images gives interesting error-correcting properties for VT and Helberg codes. If two Naisargik images of VT code generates an intersecting one deletion sphere, then the images holds the same weights. A quaternary Helberg code designed to correct $s$ deletions can effectively rectify $s+1$ deletion errors when considering its Naisargik image, and $s$-deletion correcting binary Helberg code can corrects $\lfloor\frac{s}{2}\rfloor$ errors with inverse Naisargik image.

On Naisargik Images of Varshamov-Tenengolts and Helberg Codes

TL;DR

The article investigates how Naisargik maps from to affect binary and quaternary VT and Helberg codes, with eight maps used for VT and a single map for Helberg. It shows that VT images generally fail to preserve unit deletion-correcting properties, while Helberg codes gain increased deletion-correcting power under these maps (and their inverses). The work provides conjectures, partial proofs, and extensive data suggesting structured relations between codewords across alphabets, including a proposed one-to-one correspondence between certain quaternary and binary Helberg code families. These findings have implications for designing robust deletion/insertion-correcting codes for DNA storage and related channels, and they open paths for extending mappings to broader -ary settings.

Abstract

The VT and Helberg codes, both in binary and non-binary forms, stand as elegant solutions for rectifying insertion and deletion errors. In this paper we consider the quaternary versions of these codes. It is well known that many optimal binary non-linear codes like Kerdock and Prepreta can be depicted as Gray images (isometry) of codes defined over . Thus a natural question arises: Can we find similar maps between quaternary and binary spaces which gives interesting properties when applied to the VT and Helberg codes. We found several such maps called Naisargik (natural) maps and we study the images of quaternary VT and Helberg codes under these maps. Naisargik and inverse Naisargik images gives interesting error-correcting properties for VT and Helberg codes. If two Naisargik images of VT code generates an intersecting one deletion sphere, then the images holds the same weights. A quaternary Helberg code designed to correct deletions can effectively rectify deletion errors when considering its Naisargik image, and -deletion correcting binary Helberg code can corrects errors with inverse Naisargik image.
Paper Structure (15 sections, 12 theorems, 126 equations, 18 tables, 1 algorithm)

This paper contains 15 sections, 12 theorems, 126 equations, 18 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Naisargik Maps for ${\mathbb{Z}}_4\rightarrow {\mathbb{Z}}_2^2$ and Codes
  4. Varshamov-Tenengolts Code (VT Code)
  5. Naisargik Image of Quaternary VT Code
  6. Error Correction For Naisargik Image of Quaternary VT Code
  7. Non-binary Helberg codes
  8. Naisargik Image of Quaternary Helberg Code
  9. Inverse Naisargik Image Of Binary Helberg Code
  10. Reduction Code of Helberg Code
  11. Torsion code of Helberg Code
  12. Asymptotic Bounds and Cardinality Comparison
  13. Conclusion
  14. Data Availability
  15. Appendix

Key Result

Lemma 1

For $\phi = \phi_8$ from Table tab:mapping, for any $x \in {\mathbb{Z}}_4^n$ and $X \in {\mathbb{Z}}_2^{2n}$ if $X = \phi(x)$, then where $x_i$ represents $i^{th}$ bit of sequence x.

Theorems & Definitions (37)

  • Definition 1: Binary VT Code:
  • Definition 2: q-ary VT Code
  • Example 1
  • Lemma 1
  • proof
  • Example 2
  • Lemma 2
  • proof
  • Conjecture 1
  • Proposition 1
  • ...and 27 more