Infinite Families of Few Weight Optimal Binary Linear Codes from Multivariable Functions

We study the binary linear code families associated with certain types of multivariable functions. We observe that a majority of these codes are not optimal codes nor few weight codes yet. In this paper, we find infinite families of few weight (near-) optimal binary linear codes from our code families. Furthermore, we produce support t-designs (t = 2 or 3) which cannot be determined by the Assmus-Mattson Theorem; this is the first time that the result by Tang et al. was successfully used to prove that linear codes hold t-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using the CSS construction. As a main method, we use the modified shortening method (simply, called shortening method), which is applied to our code families. Using the results on the weight distributions of our shortened codes, we verify that our codes families support t-designs (t = 2, 3). We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.