A product construction for perfect codes over arbitrary alphabets (Corresp.)

A general product construction for perfect single-error-correcting codes over an arbitrary alphabet is presented. Given perfect single-error-correcting codes of lengths
n, m
, and
q + 1
over an alphabet of order
q
, one can construct perfect single-error-correcting codes of length
(q - 1)nm + n + m
over the same alphabet. Moreover, if there exists a perfect single-error-correcting code of length
q + 1
over an alphabet of order
q
, then there exist perfect single-error-correcting codes of length
n
,
n = (q^{t} _ 1)/(q - 1)
, and
(t > 0
, an integer). Finally, connections between projective planes of order
q
and perfect codes of length
q + 1
over an alphabet of order
q
are discussed.