An investigation is made of the family of codes which are supercodes of the first-order and subcodes of the second-order Reed-Muller codes. These codes are in a one-to-one correspondence with subsets of alternating bilinear forms and it is shown how their distance enumerators can be obtained. A nice duality relation is defined on the set of linear codes in this family which relates their weight enumerators. The best codes of this family are defined and constructed, most of which are new nonlinear codes. The main part of the paper is devoted to a proof of the properties of the "dual" families of nonlinear codes which were announced in a letter by Goethals (1974). © 1976 Academic Press, Inc.
Goethals, J. M. (1976). Nonlinear codes defined by quadratic forms over GF(2). Information and Control, 31(1), 43–74. https://doi.org/10.1016/S0019-9958(76)90384-3