Coding techniques for correcting combinations of deletion, insertion and substitution errors are implemented in novel data storage applications. Error correcting codes have been well-studied for either substitution errors, or deletion and insertion (indel) errors, but the understanding of codes that correct combinations of these errors falls short. To achieve an efficient implementation in terms of redundancy, we are interested in the maximal size of a code that can correct t indels and s substitutions. Determining this maximal size in general is a difficult task, and thus in many instances we have to rely on bounds. In this thesis, we review existing bounds on the maximal size of both t-indel correcting codes and s-substitution correcting codes. Thereafter, we study how these bounds can be generalized to the setting of t-indel s-substitution correcting codes.  The main contributions of this thesis include two new explicit lower bounds, which are based on generalizations of the Gilbert-Varshamov bound. Furthermore, we show that the Singleton upper bound, several existing sphere-packing upper bounds and an upper bound based on matchings in hypergraphs can all be generalized to upper bounds on the maximal size of t-indel s-substitution correcting codes. Several of these bounds provide improvements upon existing results in literature. Moreover, we argue that the asymptotic redundancy of maximally sized t-indel s-substitution correcting codes lies between (t + s) logq(n) and 2(t + s) logq(n) + o(logq(n)).

For many IT applications information is stored digitally across multiple storage units and needs to be available continuously. Due to malfunctions data servers might be erased or temporarily inaccessible. Different approaches using linear coding theory have been proposed in order to retrieve the content of erased servers from repair sets containing a selection of the remaining servers. This thesis focuses on comparing various repair methods for two erasures; disjoint parallel repair which protects each erasure by multiple disjoint repair sets, cooperative repair which uses a single repair set to repair all erasures at once and sequential repair which repairs erasures individually by using already repaired erasures. Coding inevitably induces storage overhead measured by the information rate. The applied method creates transmission overhead measured by locality, which concerns the number of servers accessed to perform erasure repair. The comparison mainly consists of appropriately computing minimal transmission overhead for these methods given a predetermined storage overhead. It is shown that Hamming codes have the best possible transmission overhead applying the cooperative method. The next best method is sequential repair, whereas disjoint parallel repair is too restrictive for two erasure repair with Hamming codes. For a generalized parity code it is demonstrated that the sequential method can repair more erasures than the disjoint parallel approach, and reach a better locality than the cooperative method. In general, the cooperative method eciently uses access to servers in order to reduce transmission overhead. For the same purpose, the sequential method uses already repaired servers allowing smaller repair sets to be accessed. In any repair process it is key to nd optimal combinations of a method and code which exploit these qualities. The cooperative method applied to Hamming codes and the sequential method combined with generalized parity codes prove to be high-ranking combinations in this regard.