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 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 PL, respectively. The code PL 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 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 ℤ4-analog of the first-order Reed-Muller code. As a result, PL has a simple description in the ℤ4-domain that admits a simple syndrome decoder. At length 16, the code PL coincides with the Preparata code.

Duke Scholars

Author Robert Calderbank Computer Science