Locally Decodable Code (LDC) is a code that encodes a message in a way that one can decode any particular symbol of the message by reading only a constant number of locations, even if a constant fraction of the encoded message is adversarially corrupted.

In this paper we present a new approach for the construction of LDCs. We show that if there exists an irreducible representation (V) of G and q elements g1g2gq in G such that there exists a linear combination of matrices (gi) that is of rank one, then we can construct a q-query Locally Decodable CodeC:V\FG.

We show the potential of this approach by constructing constant query LDCs of sub-exponential length matching the parameters of the best known constructions