Abstarct: GQC codes are usually discussed in terms of submodules over polynomial rings. The generator matrix and parity-check matrix of a GQC code viewed as a linear code can be represented by a combination of circulant matrices with different numbers of columns. Similar to QC codes, a GQC code can also be represented by a polynomial matrix. Generalized integer codes are defined as codes over rings of integers modulo n in which individual code symbols generally have different moduil. In this thesis, we use a certain type of matrix identities to derive a necessary and sufficient condition for integer matrices to be equal to the generator matrices of generalized integer codes. Moreover, it is shown that the parity check matrix is generated from this matrix identity of the generator matrix. Finally, an efficient algorithms which calculates all generator polynomial matrices of Generalized integer codes and also enumerates theoretically all of the generator matrices of generalized integer codes is provided. Keywords: Codes over rings, Generalized integer code, Generalized quasi-cyclic code, ‎Buchbergers algorithm, Generator matrix, Parity check matrix, Polynomial matrix.