Group Codes with Low-Density Orthogonal Idempotent

We introduce the family of Low-Density Orthogonal Idempotent (LDOI) codes, which are group codes characterized by two-sided ideals of a semisimple group algebra that have an orthogonal idempotent with low Hamming weight. These codes can be thought of as the analog, over a group algebra, of Low-Density Parity-Check (LDPC) codes over finite fields. We initiate the study of LDOI codes and characterize some of their properties in terms of weight of the orthogonal idempotent and the so-called adjacency matrix. We then show how the iterative Bit Flipping (BF) algorithm-the simplest form of decoder used for LDPC codes-can be adapted to decode LDOI codes. We show that, for certain families of LDOI codes (namely, those having a binary adjacency matrix), the BF decoder is optimal (i.e., achieves maximum error correction capability) even when just one iteration is performed.