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.
Wee, van, G. J. M., Cohen, G. D., & Litsyn, S. N. (1991). A note on perfect multiple coverings of the Hamming space. IEEE Transactions on Information Theory, 37(3), 678-682. https://doi.org/10.1109/18.79931