Codes and Stability

Abstract: We introduce new yet easily accessible codes for elements of $GL_r(A)$ with $A$ the adelic ring of a (dimension one) function field over a finite field. They are linear codes, and coincide with classical algebraic geometry codes when $r=1$. Basic properties of these codes are presented. In particular, when offering better bounds for the associated dimensions, naturally introduced is the well-known stability condition. This condition is further used to determine the minimal distances of these codes. To end this paper, for reader's convenience, we add two appendices on some details of the adelic theory of curves and classical AG codes, respectively.