Selforthogonality of $q$ary Images of $q^m$ary Codes and Quantum Code Construction
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 selforthogonality 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 selforthogonal. The conditions require that the original code be selforthogonal 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 selforthogonal codes result in selforthogonal images. Another result is that image of a code is selforthogonal for all bases if and only if trace of the code is selforthogonal, except for the case of binary images of 4ary codes. The conditions are particularly simple to state and apply for cyclic codes. To illustrate a possible application, new quantum errorcorrecting codes have been constructed with larger minimum distance than previously known.
 Publication:

arXiv eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:cs/0606106
 Bibcode:
 2006cs........6106B
 Keywords:

 Computer Science  Information Theory
 EPrint:
 submitted to IEEE Transactions on Information Theory