Mathematical coding theory addresses the problem of transmitting information reliably and efficiently across noisy channels. Low-density parity-check (LDPC) codes are graph-based codes whose representation, encoding, and/or decoding can be visualized using a sparse graph representation, called a Tanner graph. The majority of this talk will focus on absorbing sets, which are combinatorial structures in a code’s Tanner graph that have been shown to characterize iterative decoder failure of LDPC codes. Previously, absorbing sets have been regarded as combinatorial, graph-theoretic objects, but in this talk, I demonstrate a novel algebraic connection between absorbing sets and the cosets and syndromes of their support vectors. Time permitting, I will also give highlights of some of my current projects in graph-based codes.