On $[[n,n-4,3]]_{q}$ Quantum MDS Codes for odd prime power $q$

TL;DR

This paper constructs new quantum MDS codes over finite fields for odd prime powers by leveraging Hermitian self-orthogonal codes, expanding the range of known quantum codes with optimal parameters.

Contribution

It introduces a method to construct quantum MDS codes for odd prime powers using Hermitian self-orthogonal codes over finite fields, covering a broad range of lengths.

Findings

Constructed quantum MDS codes for 4 ≤ n ≤ q^2 + 1

Utilized Hermitian self-orthogonal codes over GF(q^2)

Achieved codes with parameters [[n, n-4, 3]]_q

Abstract

For each odd prime power $q$, let $4 \leq n\leq q^{2}+1$. Hermitian self-orthogonal $[n,2,n-1]$ codes over $GF(q^{2})$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n-4,3]]_{q}$ quantum MDS codes for $4 \leq n\leq q^{2}+1$ are obtained.