On the apparent duality of the Kerdock and Preparata codes

Abstract

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 P _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 ℤ₄ of integers modulo 4. We exhibit a quaternary code Q and its quaternary dual P ^⊥ which, under the Gray mapping, give rise to the Kerdock code K, and Preparata-like code P _L, respectively. The code P _L is identical in weight and distance distribution to the extended Preparata code. The linearity of Q and P ^⊥ ensures that the binary codes K and P _L are distance invariant, while their duality as quaternary codes guarantees that K and P _L have dual weight distributions. The quaternary code Q is the ℤ₄-analog of the first-order Reed-Muller code. As a result, P _L has a simple description in the ℤ₄-domain that admits a simple syndrome decoder. At length 16, the code P _L coincides with the Preparata code.

This work was supported in part by the National Science Foundation under Grant NCR-9016077 and by Hughes Aircraft Company under its Ph.D. fellowship program.