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Linear codes with few weights over $\mathbb{F}_{p}+u\mathbb{F}_{p}$

Pavan Kumar, Noor Mohammad Khan

Abstract

For any positive integer $m$ and an odd prime $p$; let $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $q=p^{m}$, be a ring extension of the ring $\mathbb{F}_{p}+u\mathbb{F}_{p}.$ In this paper, we construct linear codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$ by using trace function defined on $\mathbb{F}_{q}+u\mathbb{F}_{q}$ and determine their Hamming weight distributions by employing symplectic-weight distributions of their Gray images.

Linear codes with few weights over $\mathbb{F}_{p}+u\mathbb{F}_{p}$

Abstract

For any positive integer and an odd prime ; let , where , be a ring extension of the ring In this paper, we construct linear codes over by using trace function defined on and determine their Hamming weight distributions by employing symplectic-weight distributions of their Gray images.
Paper Structure (5 sections, 19 theorems, 68 equations)

This paper contains 5 sections, 19 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Primary results needed to establish main results
  4. Main results
  5. Conclusions

Key Result

Lemma 2.1

LN97 Let the symbols have the same meanings as before. Then $G(\eta, \chi_{1})=(-1)^{m-1}\sqrt{-1}^{\frac{(p-1)^{2}m}{4}}\sqrt{p^m},~G(\overline{\eta}, \overline{\chi}_{1})=\sqrt{-1}^{\frac{(p-1)^{2}}{4}}\sqrt{p}.$

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 13 more