  Code-based Distributed Matrix Multiplication (DMM) has been extensively
studied in distributed computing for efficiently performing large-scale matrix
multiplication using coding theoretic techniques. The communication cost and
recovery threshold (i.e., the least number of successful worker nodes required
to recover the product of two matrices) are two major challenges in coded DMM
research. Several constructions based on Reed-Solomon (RS) codes are known,
including Polynomial codes, MatDot codes, and PolyDot codes. However, these
RS-based schemes are not efficient for small finite fields because the
distributed order (i.e., the total number of worker nodes) is limited by the
size of the underlying finite field. Algebraic geometry (AG) codes can have a
code length exceeding the size of the finite field, which helps solve this
problem. Some work has been done to generalize Polynomial and MatDot codes to
AG codes, but the generalization of PolyDot codes to AGcodes still remains an
open problem as far as we know. This is because functions of an algebraic curve
do not behave as nicely as polynomials.
  In this work, by using local expansions of functions, we are able to
generalize the three DMM schemes based on RS codes to AG codes. Specifically,
we provide a construction of AG-based PolyDot codes for the first time. In
addition, our AG-based Polynomial and MatDot codes achieve better recovery
thresholds compared to previous AG-based DMM schemes while maintaining similar
communication costs. Our constructions are based on a novel basis of the
Riemann-Roch space using local expansions, which naturally generalizes the
standard monomial basis of the univariate polynomial space in RS codes. In
contrast, previous work used the non-gap numbers to construct a basis of the
Riemann-Roch space, which can cause cancellation problems that prevent the
conditions of PolyDot codes from being satisfied.
