A cyclically permutable code is a binary code the codewords of which are cyclically distinct and have full cyclic order. The purpose of this paper is to construct cyclically permutable codes by means of linear p-ary constacyclic codes, where p denotes a prime number. An efficient method is used to select codewords from a constacyclic code which are constacyclically distinct and have full constacyclic order. Through a cyclic representation for the elements of GF(p) and by a correspondence of two-dimensional arrays and N-tuples, the codewords selected produce the codebook of a cyclically permutable code. The cyclically permutable codes constructed are used in the construction of protocol sequences for the M-active-out-of-T users collision channel without feedback.