  Revisiting an approach by Conway and Sloane we investigate a collection of
optimal non-linear binary codes and represent them as (non-linear) codes over
Z4. The Fourier transform will be used in order to analyze these codes, which
leads to a new algebraic representation involving subgroups of the group of
units in a certain ring.
  One of our results is a new representation of Best's (10, 40, 4) code as a
coset of a subgroup in the group of invertible elements of the group ring
Z4[Z5]. This yields a particularly simple algebraic decoding algorithm for this
code.
  The technique at hand is further applied to analyze Julin's (12, 144, 4) code
and the (12, 24, 12) Hadamard code. It can also be used in order to construct a
(non-optimal) binary (14, 56, 6) code.
