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 <italic>few weight</italic> (near-) <italic>optimal</italic> binary linear codes from our code families. Furthermore, we produce support <italic>t</italic>-designs (<italic>t</italic> = 2 or 3) which cannot be determined by the <italic>Assmus-Mattson Theorem</italic>; this is the first time that the result by Tang et al. was successfully used to prove that linear codes hold <italic>t</italic>-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using the <italic>CSS construction</italic>. As a main method, we use the <italic>modified shortening method</italic> (simply, called <italic>shortening method</italic>), 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 <italic>t</italic>-designs (<italic>t</italic> = 2, 3).We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.