A generalization of multi-twisted codes over finite fields, their Galois duals and Type II codes

Let F-q denote the finite field of order q, and let Lambda = (lambda(1), lambda(2), ..., lambda(e)), where lambda(1), lambda(2), ..., lambda(e) are non-zero elements of F-q. Let n = m(1) + m(2) + center dot center dot center dot + m(e), where m(1), m(2), center dot center dot center dot, m(e) are arbitrary positive integers (not necessarily coprime to q). In this paper, we study algebraic structures of Lambda-multi-twisted (Lambda-MT) codes of length n and block lengths (m(1), m(2), ..., m(e)) over F-q and their Galois duals (i.e., their orthogonal complements with respect to the Galois inner product on Fn q). We develop generator theory for Lambda-MT codes of length n over F-q and show that each Lambda-MT code of length n over F-q has a unique nice normalized generating set. With the help of a normalized generating set, we explicitly determine the dimension and a generating set of the Galois dual of each Lambda-MT code of length n over F-q. We also provide a trace description of all Lambda-MT codes of length n over F-q by using the generalized discrete Fourier transform (GDFT), which gives rise to a method to construct these codes. We further provide necessary and sufficient conditions under which a Euclidean self-dual Lambda-MT code of length n over F-2(e) is a Type II code when lambda i = 1 and mi = n(i)2(a) for 1 <= i <= l, where a >= 0 is an integer and n(1), n(2), ..., n(l) are odd positive integers satisfying n(1) = n(2) = ... = n(e) (mod 4). Besides this, we obtain several linear codes with best-known and optimal parameters from 1-generator Lambda-MT codes over Fq, where 2 = q = 7. It is worth mentioning that these code parameters can not be attained by any of their subclasses (such as constacyclic and quasi-twisted codes) containing record breaker codes.