One - Generator Quasi - Cyclic Quaternary Linear Codes and Construction X

Let GF (q) denote the Galois field of q elements, and let V (n, q) denote the vector space of all ordered n-tuples over GF (q). The number of nonzero positions in a vector x ∈ V (n, q) is called the Hamming weight wt(x) of x. The Hamming distance d(x,y) between two vectors x,y ∈ V (n, q) is defined by d(x,y) = wt(x − y). A linear code C of length n and dimension k over GF (q) is a k-dimensional subspace of V (n, q). The minimum distance of a linear code C is d(C) = min {d(x,y)|x,y ∈ C,x = y}. Such a code is called an [n, k, d]q code if its minimum Hamming distance is d. For a linear code, the minimum distance is equal to the smallest of the weights of the nonzero codewords. A m×m matrix B each row of which is a cyclic shift of the previous one is called a circulant matrix. A code is called p-quasi-cyclic (p-QC for short) if every cyclic shift of a codeword by p places is again a codeword. A quasi-cyclic (QC) code is just a code of length n which is p-QC for some divisor p of n with p < n [4]. A cyclic code is just a 1-QC code. Suppose C is an p-QC [pm,k]-code. It is convenient to take the co-ordinate places of C in the order