On the apparent duality of the Kerdock and Preparata codes

Abstract

The Kerdock and Preparata codes are something of an enigma in coding theory since they are both Hamming distance invariant and have weight enumerators that are dual under the MacWilliams transform just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparata-like code P/sub L/ from the Kerdock code K, why the existence of a distance-invariant code with weight distribution that is the McWilliams transform of that of the Kerdock code is only to be expected. The construction involves quaternary codes over the ring Z/sub 4/ of integers modulo 4. We exhibit a quaternary code Q and its quaternary dual Q which, under the Gray map, give rise to the Kerdock code K and Preparata-like code P/sub L/, respectively. The code P/sub L/ is identical in weight and distance distribution to the Preparata code. The linearity of Q and Q ensures that K and P/sub L/ are distance invariant, while their duality as quaternary codes guarantees that K and P/sub L/ have dual weight distributions.