Multi-twisted additive codes over finite fields and their dual codes

Sandeep Sharma (Tel Aviv University)

Additive codes over the finite field F4 were introduced and studied by Calderbank et al. (1998) as a natural generalization of linear codes. Later, Rains (1999) and Bierbrauer and Edel (2000) defined and studied additive codes over arbitrary finite fields. These codes constitute an important family of error-correcting codes and are also helpful in constructing quantum error-correcting codes. In this talk, we will introduce a new class of additive codes over finite fields: multi-twisted (MT) additive codes. We will study their algebraic structures by writing a canonical form decomposition for these codes and providing an enumeration formula. By placing ordinary, Hermitian, and * trace bilinear forms, we will further study their dual codes and derive necessary and sufficient conditions under which an MT additive code is self-dual, self-orthogonal, and an additive code with complementary dual (ACD). We will also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field and provide explicit enumeration formulae for self-dual, self-orthogonal, and ACD codes over finite fields with respect to the aforementioned trace bilinear forms. Further, using probabilistic methods, we will prove that MT additive codes and their three subclasses of self-orthogonal, self-dual, and ACD codes are asymptotically good.

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