On the Apparent Duality of the Kerdock and Preparata Codes

The Kerdock and extended Preparata codes are something of an enigma in coding theory since they are both Hamming-distance invariant and have weight enumerators that are MacWilliams duals just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparata-like code PL 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 Z4 of integers modulo 4. We exhibit a quaternary code Q and its quater-nary dual Q ? which, under the Gray mapping, give rise to the Kerdock code K and Preparata-like code PL, respectively. The code PL is identical in weight and distance distribution to the extended Preparata code. The linearity of Q and Q ? ensures that the binary codes K and PL are distance invariant, while their duality as quaternary codes guarantees that K and PL have dual weight distributions. The quaternary code Q is the Z Z4-analog of the rst-order Reed-Muller code. As a result, PL has a simple description in the Z Z4-domain that admits a simple syndrome decoder. At length 16, the code PL coincides with the Preparata code.