Codes from curves over finite fields were first developed in the late 1970's by V. D. Goppa and are known as algebraic geometry codes. Since that time, the construction has been tailored to fit particular applications, such as erasure recovery and error correction using less received information than in the classical case. The Hermitian-lifted code construction of L'opez, Malmskog, Matthews, Piñero-González, and Wootters (2021) provides codes from the Hermitian curve over $F_{q^2}$ which have the same locality as the well-known one-point Hermitian codes but with a rate bounded below by a positive constant independent of the field size. However, obtaining explicit expressions for the code is challenging. In this dissertation, we consider codes from norm-trace curves, which are a generalization of the Hermitian curve. We develop norm-trace-lifted codes and demonstrate an explicit basis of the codes. We then consider fractional decoding of codes from norm-trace curves, extending the results obtained for codes from the Hermitian curve by Matthews, Murphy, and Santos (2021). / Doctor of Philosophy / Coding theory focuses on recovering information, whether that data is corrupted and changed (called an error) or is simply lost (called an erasure). Classical codes achieve this goal by accessing all received symbols. Because long codes, meaning those with many symbols, are common in applications, it is useful for codes to be able to correct errors and recover erasures by accessing less information than classical codes allow. That is the focus of this dissertation. Codes with locality are designed for erasure recovery using fewer symbols than in the classical case. Such codes are said to have locality $r$ and availability $s$ if each symbol can be recovered from $s$ disjoint sets of $r$ other symbols. Algebraic curves, such as the Hermitian curve or the more general norm-trace curves, offer a natural structure for designing codes with locality. This is done by considering lines intersected with the curve to form repair groups, which are sets of $r+1$ points where the information from one point can be recovered using the rest of the points in the repair group. An error correction method which uses less data than the classical case is that of fractional decoding. Fractional decoding takes advantage of algebraic properties of the field trace to correct errors by downloading only a $lambda$-proportion of the received information, where $lambda < 1$. In this work, we consider a new family of codes resulting from norm-trace curves, and study their locality and availability, as well as apply the ideas of fractional decoding to these codes.

In this thesis we develop exact repair schemes capable of repairing or circumventing unavailable servers of a distributed network in the context of distributed storage and secure distributed matrix multiplication. We develop the (Λ, Γ, W, ⊙)-exact repair scheme framework for discussing both of these contexts and develop a multitude of explicit exact repair schemes utilizing decreasing monomial-Cartesian codes (DMC codes). Specifically, we construct novel DMC codes in the form of augmented Cartesian codes and rectangular monomial-Cartesian codes, as well as design exact repair schemes utilizing these constructions inspired by the schemes from Guruswami and Wootters [16] and Chen and Zhang [6]. In the context of distributed storage we demonstrate the existence of both high rate and low bandwidth systems based on these schemes, and we develop two methods to extend them to the l-erasure case. Additionally, we develop a family of hybrid schemes capable of attaining high rates, low bandwidths, and a balance in between which proves to be competitive compared to existing schemes. In the context of secure distributed matrix multiplication we develop similarly impactful schemes which have very competitive communication costs. We also construct an encoding algorithm based on multivariate interpolation and prove it is T-secure. / Doctor of Philosophy / Distributed networks may be thought of as networks of computers and/or servers which are capable of transmitting and receiving data from one another. For many applications it is possible for distributed networks to perform better than the sum of their constituent parts. In this thesis we will focus on the particular applications of distributed storage and secure distributed multiplication. A distributed storage system is a system that is capable of storing a single data file over every server in a distributed network. Distributed storage systems often come with exact repair schemes which are algorithms designed to reconstruct the data from a server in the network given the data from the other servers. In particular, if a server on the network ever fails or is otherwise unavailable an exact repair scheme can be used to repair the lost data from the server and maintain the original file. A distributed matrix multiplication scheme on the other hand is a process by which two matrices stored on a source server can be multiplied using a distributed network of helper servers. Again if a helper server becomes unavailable during this process we may use an exact repair scheme to circumvent this delay. The main goal of this thesis is to develop exact repair schemes for the distributed storage and secure distributed matrix multiplication contexts utilizing a mathematical object known as an evaluation code. We will develop several families of exact repair schemes which may be finely tuned to fit particular situations within these contexts, and we will compare these schemes to the existing schemes in the field.