A New Method for Constructing Self-Dual Codes over Finite Commutative Rings with Characteristic 2

In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for kx2k matrices M over R satisfying the conditions that its rows are linearly independent over R and MM inverted perpendicular=alpha inverted perpendicular alpha for an R-linearly independent vector alpha is an element of Rk. Let C be a linear code generated by such a matrix M. We prove that the dual code C perpendicular to of C is also a free linear code with dimension k, as well as C/Hull(C) and C perpendicular to/Hull(C) are one-dimensional free R-modules, where Hull(C) represents the hull of C. Based on these facts, an isometry from Rx+Ry onto R2 is established, assuming that x+Hull(C) and y+Hull(C) are bases for C/Hull(C) and C perpendicular to/Hull(C) over R, respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix MM inverted perpendicular takes the form of alpha inverted perpendicular alpha with alpha being a linearly independent vector in Rk, a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy MM inverted perpendicular=0. The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.