On generalized Hamming weights of \(\text{BCH}(2)^\perp\) (Q1266998)

The main result of this paper is a formula for the third generalized Hamming weight of a dual binary BCH code of length \(2^m-1\), where \(m=2^rs\), \(r>0\) and \(s\) is odd, or \(m=2^r\), \(r>5\). Also given is a simple proof of a formula of Chung and Shim for the second Hamming weight (with a restriction on \(m\)), cf. \textit{H. Chung} [Applied algebra, algebraic algorithms and error-correcting codes, Proc. 9th Int. Symp., AAECC-9, New Orleans/LA (USA) 1991, Lect. Notes Comput. Sci. 539, 118-129 (1991; Zbl 0767.94013)] and \textit{C. Shim} and \textit{H. Chung} [IEEE Trans. Inf. Theory 41, No.3, 805-808 (1995; Zbl 0820.94020)]. The methods used, which involve elliptic curves and quadratic forms over fields of characteristic 2, elucidate and extend those of Chung and Shim.