Stabiliser codes over fields of even order

TL;DR

This work shows that stabiliser codes over fields of even order $q=2^h$ are in bijection with binary stabiliser codes on $hn$ qubits, via the isomorphism ${ m GF}(2^h) o{ m GF}(2)^h$ and trace-orthogonal bases. It provides a geometric description in terms of quantum sets of symplectic polar spaces and corresponding quantum sets of lines and sets of lines, enabling transfer of code properties and equivalence through projective and symplectic transformations. The framework yields concrete results for ${ m GF}(4)$ codes, proving the unique $[[4,0,3]]_4$ code and the nonexistence of $[[7,1,4]]_4$ and $[[8,0,5]]_4$ codes, with a computational approach corroborating the classifications. The work bridges stabiliser code theory with finite geometry, offering a robust toolkit for constructing and ruling out codes via geometric configurations and providing a foundation for extensions to other characteristics and algebraic settings.

Abstract

We prove that the natural isomorphism between GF(2^h) and GF(2)^h induces a bijection between stabiliser codes on n quqits with local dimension q=2^h and binary stabiliser codes on hn qubits. This allows us to describe these codes geometrically: a stabiliser code over a field of even order corresponds to a so-called quantum set of symplectic polar spaces. Moreover, equivalent stabiliser codes have a similar geometry, which can be used to prove the uniqueness of a [[4,0,3]]_4 stabiliser code and the nonexistence of both a [[7,1,4]]_4 and an [[8,0,5]]_4 stabiliser code.

Figures (5)

• Figure 1: Operations that give an equivalent stabiliser matrix, where $L$ is invertible and each $2h\times2h$ block matrix $R_iS_iT_iU_i$ is a symplectic matrix.
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Theorems & Definitions (23)

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