Self-orthogonality of $q$-ary Images of $q^m$-ary Codes and Quantum Code Construction

TL;DR

This paper investigates conditions under which the $q$-ary image of a $q^m$-ary code is self-orthogonal, providing criteria that relate the properties of the original code, the basis, and biadditive forms, with applications to quantum error correction.

Contribution

It derives necessary and sufficient conditions for the self-orthogonality of code images over field extensions, generalizing previous results and enabling new quantum code constructions.

Findings

Self-orthogonality of images depends on the original code and basis properties.

For characteristic two fields, only self-orthogonal codes produce self-orthogonal images.

New quantum error-correcting codes with larger minimum distance were constructed.

Abstract

A code over GF$(q^m)$ can be imaged or expanded into a code over GF$(q)$ using a basis for the extension field over the base field. The properties of such an image depend on the original code and the basis chosen for imaging. Problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory. In this work, a generalized version of the problem of self-orthogonality of the $q$-ary image of a $q^m$-ary code has been considered. Given an inner product (more generally, a biadditive form), necessary and sufficient conditions have been derived for a code over a field extension and an expansion basis so that an image of that code is self-orthogonal. The conditions require that the original code be self-orthogonal with respect to several related biadditive forms whenever certain power sums of the dual basis elements do not vanish. Numerous interesting corollaries have been derived by specializing the general conditions. An interesting result for the canonical or regular inner product in fields of characteristic two is that only self-orthogonal codes result in self-orthogonal images. Another result is that image of a code is self-orthogonal for all bases if and only if trace of the code is self-orthogonal, except for the case of binary images of 4-ary codes. The conditions are particularly simple to state and apply for cyclic codes. To illustrate a possible application, new quantum error-correcting codes have been constructed with larger minimum distance than previously known.