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MDS Stabilizer Poset Codes

Mahir Bilen Can

TL;DR

A Singleton-type bound for stabilizer poset codes is established and the notion of MDS stabilizer poset codes is introduced, thereby extending classical results on MDS poset codes to the quantum setting.

Abstract

Poset metrics in the context of stabilizer codes are investigated. MDS stabilizer poset codes are defined. Various characterizations of these quantum codes are found. Methods for producing examples are proposed.

MDS Stabilizer Poset Codes

TL;DR

A Singleton-type bound for stabilizer poset codes is established and the notion of MDS stabilizer poset codes is introduced, thereby extending classical results on MDS poset codes to the quantum setting.

Abstract

Poset metrics in the context of stabilizer codes are investigated. MDS stabilizer poset codes are defined. Various characterizations of these quantum codes are found. Methods for producing examples are proposed.

Paper Structure

This paper contains 10 sections, 13 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Classical codes.
  4. Stabilizer codes.
  5. Poset metrics.
  6. Additive Poset Codes
  7. Transfer
  8. Singleton's Bound For Stabilizer Poset Codes
  9. MDS Property
  10. Final Remarks

Key Result

Theorem 1.1

Let $\mathtt{P}$ be a poset on $[n]$. If a stabilizer $\mathtt{P}$-code $Q$ has parameters $[[n,K,\mathtt{d}_{\mathtt{P}}]]_q$, where $K>1$, then the following inequality holds: $K \leq q^{n-2\mathtt{d}_{\mathtt{P}}+2}$.

Figures (1)

  • Figure 1: The Hasse diagram of a poset on $[8]$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.1
  • Theorem 1.5
  • Example 2.2
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 26 more