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Stabilizer quantum codes defined by trace-depending polynomials

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz, Diego Ruano

TL;DR

This paper is able to provide new binary records and non-binary codes improving the ones available in the literature and gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters.

Abstract

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters. Namely, we are able to provide new binary records and non-binary codes improving the ones available in the literature.

Stabilizer quantum codes defined by trace-depending polynomials

TL;DR

This paper is able to provide new binary records and non-binary codes improving the ones available in the literature and gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters.

Abstract

Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters. Namely, we are able to provide new binary records and non-binary codes improving the ones available in the literature.
Paper Structure (10 sections, 15 theorems, 100 equations, 13 tables)

This paper contains 10 sections, 15 theorems, 100 equations, 13 tables.

Table of Contents

  1. Introduction
  2. Evaluation codes and $b$th trace-depending polynomials
  3. The $b$th trace-depending polynomials
  4. Evaluation codes
  5. Stabilizer quantum codes from subfield-subcodes. Examples
  6. Subfield-subcodes
  7. Examples
  8. Binary examples
  9. Non-binary examples
  10. Sporadic codes from trace-depending polynomials

Key Result

Proposition 3

The degree of the $b$th trace-depending polynomial $\mathrm{Tr}_b(X)$, $b=1 +q^t, \; 1 < t \leq n$, is $m=m(t) = q^{2n-1-t} + q^{2n-1}$.

Theorems & Definitions (30)

  • Definition 1
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 5
  • Proposition 6
  • proof
  • Remark 7
  • ...and 20 more