A note on perfect multiple coverings of the Hamming space

Let Q be an alphabet of size q>or=2. The Hamming space Qⁿ that consists of all n-tuples of elements of Q is a metric space, provided with the Hamming distance function. A perfect multiple covering (PMC) is a code C in Qⁿ such that there exist fixed numbers r and mu with the property that every word in Qⁿ is within distance r from exactly mu codewords of C. The authors give a few constructions of PMCs and investigate in detail the problem of determining all possible parameters of PMCs with r=1.