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Construction of Linear Codes from the Unit Graph $G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})$

Wajid M. Shaikh, Rupali S. Jain, B. Surendranath Reddy

TL;DR

The python code for generating unit graph G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m}) for any integers for any prime $r$ is developed and conjectures two on linear codes constructed from the unit graph are stated.

Abstract

In this paper, we develop the python code for generating unit graph $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$, for any integers $m\ \& \ n$. For any prime $r$, we construct $r$-ary linear codes from the incidence matrix of the unit graph $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$, where $n \ \& \ m$ are either power of prime or product of power of primes. We also prove the minimum distance of dual of the constructed codes as either 3 or 4. Finally, we state conjectures two on linear codes constructed from the unit graph $G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})$, for any integer $m\ \& \ n$.

Construction of Linear Codes from the Unit Graph $G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})$

TL;DR

The python code for generating unit graph G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m}) for any integers for any prime is developed and conjectures two on linear codes constructed from the unit graph are stated.

Abstract

In this paper, we develop the python code for generating unit graph , for any integers . For any prime , we construct -ary linear codes from the incidence matrix of the unit graph , where are either power of prime or product of power of primes. We also prove the minimum distance of dual of the constructed codes as either 3 or 4. Finally, we state conjectures two on linear codes constructed from the unit graph , for any integer .
Paper Structure (6 sections, 22 theorems, 7 equations, 2 figures)

This paper contains 6 sections, 22 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G(\mathbb{Z}_{n}\oplus \mathbb{Z}_{m})$ and Python Code
  4. Linear Codes from Unit Graph $G(\mathbb{Z}_{p^{n}}\oplus \mathbb{Z}_{q^{m}})$
  5. Linear Codes from the Unit Graph $G(\mathbb{Z}_{p_{1}^{n_1}p^{n_2}_{2}} \oplus \mathbb{Z}_{q_{1}^{m_1}q^{m_2}_{2}})$
  6. Conclusion

Key Result

Theorem 2.3

18 Let $C_{r}$ be a $r$-ary code of length $n$ over a field $\mathbb{F}_{r}$. Then $C^{\perp}_{r}$ is a linear code of length $n$ and $\text{dim}(C_{r}^{\perp})=n-\text{dim}(C_{r})$.

Figures (2)

  • Figure 1: Unit graph of $G(\mathbb{Z}_{5}\oplus \mathbb{Z}_{5})$
  • Figure :

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 35 more