New projective codes identified for noncommutative rings and group codes
Free and projective LCD codes
Information Theory
Summary
Coding theory studies how to protect information from errors using codes made with mathematical rings. The authors show that certain codes called LCD codes are always projective, which helps understand their structure better, especially when the rings involved are more complicated. They also find a way to determine when group-based codes are LCD by looking at special elements called central idempotents. This work extends earlier results to more complex settings.
What this means in practice
- •For coding theory engineers: Use characterisation of projective LCD codes over noncommutative rings to design robust error-correcting codes with predictable algebraic properties.
- •For cryptographic system designers: Identify LCD group codes generated by central idempotents to construct codes with desirable duality properties for secure communication.
A theory result. No direct application yet.
Authors
Dominik Krasula
Abstract
LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the size condition of a code and its dual to the noncommutative setting. This result is applied to prove that group codes are LCD if and only if they are generated by a central idempotent.