This paper presents secure MatDot codes, a family of evaluation codes that support secure distributed matrix multiplication via a careful selection of evaluation points that exploit the properties of the dual code. We show that the secure MatDot codes provide security against the user by using locally recoverable codes. These new codes complement the recently studied discrete Fourier transform codes for distributed matrix multiplication schemes that also provide security against the user. There are scenarios where the associated costs are the same for both families and instances where the secure MatDot codes offer a lower cost. In addition, the secure MatDot code provides an alternative way to handle the matrix multiplication by identifying the fastest servers in advance. In this way, it can determine a product using fewer servers, specified in advance, than the MatDot codes which achieve the optimal recovery threshold for distributed matrix multiplication schemes.

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes (DM-CC), a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

We present two families of augmented Reed-Muller (ARM) codes, which are evaluation codes obtained by adding specific vectors to a Reed-Muller code. We develop exact repair schemes for single erasures for these ARM codes. When a dimension and a base field are fixed, we give examples where ARM codes provide a lower bandwidth in comparison with Reed-Solomon codes. We analyze the asymptotical behavior when ARM codes achieve the maximum rate.