In this paper

the rate and distortion of vector quantization are discussed.It is proved that the minimum rate for encoding a source with munimum achievable distortion Dmin for a given number N of k-dimensional output codevectors is the output entropy of the optimal k-dimensional vector quantizer with codebook size N.It is also shown that for a codebook with symmetrical probability distribution

the output entropy of the vector quantizer will never be smaller than one bit per vector.Finally

a series of inequalities is derived

which gives a quantitative relation between the performance of a vector quantizer and rate-distortion function.