In this paper, we study embedding efficiency, which is an important attribute of steganographic schemes directly influencing their security. It is defined as the expected number of embedded random message bits per one embedding change. Constraining ourselves to embedding realized using linear covering codes (so called matrix embedding), we show that the quantity that determines embedding efficiency is not the covering radius but the average distance to code. We demonstrate that for linear codes of fixed block length and dimension, the highest embedding efficiency (the smallest average distance to code) is not necessarily achieved using codes with the smallest covering radius. Nevertheless, we prove that with increasing code length and fixed rate (i.e., fixed relative message length), the relative average distance to code and the relative covering radius coincide. Finally, we describe several specific examples of q-ary linear codes with q matched to the embedding operation and experimentally demonstrate the improvement in steganographic security when incorporating the coding methods to digital image steganography.
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