  Motivated by the approach of random linear codes, a new distance in the
vector space over a finite field is defined as the logarithm of the "surface
area" of a Hamming ball with radius being the corresponding Hamming distance.
It is named entropy distance because of its close relation with entropy
function. It is shown that entropy distance is a metric for a non-binary field
and a pseudometric for the binary field. The entropy distance of a linear code
is defined to be the smallest entropy distance between distinct codewords of
the code. Analogues of the Gilbert bound, the Hamming bound, and the Singleton
bound are derived for the largest size of a linear code given the length and
entropy distance of the code. Furthermore, as an important property related to
lossless joint source-channel coding, the entropy distance of a linear encoder
is defined. Very tight upper and lower bounds are obtained for the largest
entropy distance of a linear encoder with given dimensions of input and output
vector spaces.
