On monomial codes in modular group algebras

Let p be a prime number and K be the finite field of p elements, i.e. K=GF(p). Further let G be an elementary abelian p-group of order pm. Then the group algebra K[G] is modular. We consider K[G] as an ambient space and the ideals of K[G] as linear codes. A basis of a linear space is called visible, if there exists a member of the basis with the minimum (Hamming) weight of the space. The group algebra approach enables us to find some linear codes with a visible basis in the Jacobson radical of K[G]. These codes can be generated by “monomials” (Drensky & Lakatos, 1989). For p>2, some of our monomial codes have better parameters than the Generalized Reed-Muller codes. In the last part of the paper we determine the automorphism groups of some of the introduced codes.