Linear or Nonlinear?
The Z4 Origins of Kerdock and Preparata Codes

The Kerdock and Preparata codes are powerful binary, nonlinear error-correcting codes that long defied explanation. Each possesses twice as many codewords as the best-performing linear codes of the same length and minimum distance. The Preparata code, in particular, was recognized as optimal in size. Yet, despite their non-linearity, these codes exhibited a surprising behavior: they appeared to function as dual linear codes, a paradox that puzzled researchers for over a decade. So inexplicable was this behavior that one prominent theorist dismissed it as "merely a coincidence".
Professor P. V. Kumar and his co-authors resolved this long-standing mystery by showing that the Kerdock and Preparata codes are projections under a simple nonlinear map of a linear code and its dual over the ring Z4 (integers modulo 4). Their highly cited breakthrough, which earned the 1994 IEEE Information Theory Society Prize Paper Award, provided a compelling algebraic framework for understanding these exceptional codes. It also received recognition in popular science outlets such as Science magazine.
Beyond resolving the duality puzzle, this discovery sparked subsequent research into codes over rings, an activity that continues to this day. Codes over rings have since found applications in quantum error correction and DNA computing.

Reference

1. Encyclopedia of Mathematics, European Mathematical Society, Detailed page.

2. Kumar, et al., The Z4-linearity of Kerdock, Preparata, Goethals and related codes", IEEE Trans. on Information Theory, 1994 - IEEE Information Theory Society Paper Award (2050 Google Scholar Citations).

3. Barry Cipra, "Nonlinear codes straighten up and get to work", Science, Oct. 29, 1993.

4. S. Dougherty, "Algebraic Coding Theory over Finite Commutative Rings", 2017 (see p.2).