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Construction of Minimal Binary Linear Codes of dimension $n+3$

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

Abstract

In this paper, we will give the generic construction of a binary linear code of dimension $n+3$ and derive the necessary and sufficient conditions for the constructed code to be minimal. Using generic construction, a new family of minimal binary linear code will be constructed from a special class of Boolean functions violating the Ashikhmin-Barg condition. We also obtain the weight distribution of the constructed minimal binary linear code.

Construction of Minimal Binary Linear Codes of dimension $n+3$

Abstract

In this paper, we will give the generic construction of a binary linear code of dimension and derive the necessary and sufficient conditions for the constructed code to be minimal. Using generic construction, a new family of minimal binary linear code will be constructed from a special class of Boolean functions violating the Ashikhmin-Barg condition. We also obtain the weight distribution of the constructed minimal binary linear code.
Paper Structure (5 sections, 14 theorems, 98 equations, 2 tables)

This paper contains 5 sections, 14 theorems, 98 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Minimal Binary Linear Codes of dimension $n+3$ from a class of Boolean Functions
  4. A family of Minimal Binary Linear Codes of dimension $n+3$
  5. Conclusion

Key Result

Lemma 1.1

11[Ashikhmin-Barg] Let $p$ be a prime. Then a linear code $\mathscr{C}$ over $\mathbb{F}_{p}$ is minimal if where, ${wt}_{min}$ and ${wt}_{max}$ denote the minimum and maximum non-zero Hamming weights for $\mathscr{C}$, respectively.

Theorems & Definitions (27)

  • Lemma 1.1
  • Definition 2.1: $Support$
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • ...and 17 more