Equivalence of Classical and Quantum Codes

• Main Author: Pllaha, Tefjol
• Format: Others
• Published: UKnowledge 2019
• Subjects: Frobenius alphabets; isometries of codes; MacWillimas Extension Theorem; quantum stabilizer codes; LU-LC conjecture; Algebra; Computer Sciences
• Online Access: https://uknowledge.uky.edu/math_etds/59; https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1060&context=math_etds

In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.