Algebraic Structures and Metrics in Ring-Linear Coding Theory

Ring-linear coding theory extends classical coding theory by employing finite rings as alphabets rather than finite fields. Initially introduced by Assmus and Mattson in 1963, the theory gained prominence in the 1990s when connections were found between Kerdock and Preparata codes and $\mathbb{Z}_4$-linear codes. The two families behave as if they are duals of each other, with the weight enumerator of the Preparata code being the MacWilliams transform of the weight enumerator of the Kerdock code. In 1994, Hammons et al. proved that both code families can be expressed as Gray images of cyclic codes over $\mathbb{Z}_4$ in the Lee metric. Recently, ring-linear coding theory has also found promising applications. Examples include DNA storage and cryptographic applications. In particular, the need to reduce the public key size in code-based cryptosystems has led to an increased interest in exploring different ambient spaces and metrics other than vector spaces over finite fields equipped with the Hamming metric, with Lee-metric codes emerging as an alternative.\\ The first major focus of the thesis is on locally recoverable codes (LRCs) over finite chain rings. These codes enable data recovery by accessing a small subset of symbols instead of the entire codeword. We address the case of locally recoverable codes over finite chain rings. We derive a bound on the minimum distance of a locally recoverable code over a finite chain ring. We then introduce a new family of locally recoverable codes, extending a construction proposed by Tamo and Barg in 2014. We will discuss the optimality of the construction and provide some generalizations. The second central topic of this dissertation is given by the MacWilliams identities for ring-linear codes equipped with additive metrics such as the Lee, homogeneous, and subfield metrics. The thesis introduces novel partitions of the ambient space, enabling the derivation of MacWilliams-type identities where classical identities fail. We then derive a linear programming bound for the considered metrics. In addition, we derive new linear equations for the Hamming weight distribution of linear codes over finite chain rings, for which the MacWilliams identities notably hold. By leveraging the structure of parity-check matrices, we derive explicit formulas for the weight distributions of specific classes of codes, including MDR and AMDR.\\ We finally propose efficient decoding algorithms for ring-linear codes equipped with the Hamming, rank, and Lee metrics. In particular, we propose a new framework that leverages the ring structure to construct more efficient decoding algorithms for the aforementioned metrics.