Nonlinear codes defined by quadratic forms over GF(2)

Jean-Marie Goethals
1976 Information and Control  
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
more » ... r 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) . codes is that their distance enumerator is related to that of tile nonlinear codes discovered by Preparata (1968) by the MacWilliams identity, (cf. MacWilliams et aL, 1972) . A similar property holds for the infinite family of pairs of nonlinear codes C, C', described by Goethals (1974) , where C' is obtained from a (m/2-1)-optimal code and C with minimum distance 8 contains four times as many codevectors as the extended BCH code of the same length and minimum distance. The main part of this paper, Section 5, is devoted to a proof of the properties of these codes which were announced in a letter by Goethals (1974) . Finally, in Section 6, we show how easily the weight enumerators of some cyclic subcodes of the second-order Reed-Muller codes can be derived by our methods. These weight enumerators were already obtained by Kasami (1969) and Kerdock et al. (1974), but we feel that our approach was worth mentioning. QUADRATIC FORMS AND SECOND-ORDER REED--MULLER CODES The geometric nature of Reed-Muller codes is now widely known (cf. Peterson and Weldon, 1972) . Here, we shall give an account of these codes, starting from purely geometrical concepts. Let V = V(m, 2) be an m-dimensional vector space over the field B: = GF(2). (m/> 2). A linear form on V is a function L from V to ~: satisfying
doi:10.1016/s0019-9958(76)90384-3 fatcat:iu5jwneiangh7nqubeypn2tjam