Complete weight enumerators of some linear codes and their applications

Complete weight enumerators of some linear codes from quadratic forms

“…We also determined the complete weight enumerators of these q-ary linear codes. Our results extended some recent work in [9,22,27] and [18].…”

“…Remark 1 When q = p and the quadratic form Q(x) is chosen to define the sets D listed in (a)-(d), the corresponding linear codes are exactly the earlier ones documented in [9,18,27] and [21]. For the sets D listed in (a) and (c) defined by two specific Q(x) with full rank, the complete weight enumerators of C D were reported in [18,22], respectively.…”

“…It was shown that the codes from the defining sets above are two-weight or three-weight linear codes and could be optimal in certain cases [9,18,21,27]. The complete weight enumerators of the linear codes C D from D listed in (a) and (c) were determined in [22] and [18], respectively .…”

“…For the sets D listed in (a) and (c) defined by two specific Q(x) with full rank, the complete weight enumerators of C D were reported in [18,22], respectively. Our results on the complete weight enumerators work for any quadratic from over F q and thus include the results in [18,22] as special cases.…”

“…Later, Kuzmin and Nechaev obtained the complete weight enumerators of the generalized Kerdock codes and related linear codes over Galois rings [15,16]. Very recently, the complete weight enumerators of some special linear codes have been established with the theory of exponential sums over finite fields [17,18,22,23].…”