Codes in the Sum-Rank Metric and Codes over Rings

Kurzfassung

This thesis explores error-correcting codes beyond the classical framework of linear codes over finite fields in the Hamming metric. Two different directions are pursued: first, codes defined over finite rings, which enrich the algebraic framework and enable constructions not possible over fields, and second, codes in the sum-rank metric, a distance measure that combines properties of the Hamming metric and rank metric and is increasingly relevant in applications such as multishot network coding, space-time coding, code-based cryptography and distributed storage. Together, these investigations uncover new structural phenomena and establish refined performance bounds for coding theory. The first part considers cyclic left module codes over finite rings within the framework of skew polynomial rings. By transforming duality conditions from a set of differential equations into a system of multivariate polynomial equations solvable via Gröbner bases, dual-containing codes can be constructed algorithmically. The analysis reveals several special features: the dual of a cyclic module code is not necessarily cyclic, non-trivial endomorphisms and derivations are indispensable for certain constructions, and certain codes arise only in the presence of non-central modulus polynomials. These results extend the theory of codes over rings and might be of interest for the design of quantum error-correcting codes, where dual containment is essential. The second part focuses on codes in the sum-rank metric. A geometric perspective is employed by analyzing spheres, balls, and their intersections, leading to efficiently computable volume bounds. Building upon this foundation, classical results such as the Gilbert–Varshamov, sphere-packing, and Singleton bounds are generalized to the sum-rank-metric setting for linear codes, and asymptotic variants are derived. It is shown that random linear codes achieve a Gilbert–Varshamov-like bound with high probability. Covering properties are also studied, yielding upper and lower bounds on code sizes for given covering radii. Overall, this part characterizes the trade-offs between redundancy and error-correcting capability and advances the algebraic and geometric understanding of error-correcting codes in the sum-rank-metric setting.