On decoding linear error-correcting codes--I

A technique is described for finding simply computable numerical-valued functions of a received binary word whose value indicates where errors in transmission have occurred. Although it seems that a certain condition must usually be fulfilled for such functions to exist, or for our method to constitute an efficient procedure for finding them, there is, on the one hand, a strong tendency for "good" codes to satisfy the condition, while, on the other, it appears to be straightforward to construct codes which are good for a specified channel and also fulfill the condition. An advantage of the resulting decoding procedure is that it corrects and detects all possible errors; more precisely, if a word
u
is received and the coset
\bar{u}
to which
u
belongs has a unique leader
e
, the procedure concludes that
u + e
was sent, while if
u
has no unique leader, that fact, along with the weight of
\bar{u}
(and sometimes a little more) can be indicated. The ideas and techniques are illustrated by the construction of decoding procedures for the perfect (23, 12) three-error-correcting code.