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 offew weight(near-)optimalbinary linear codes from our code families. Furthermore, we produce supportt-designs (t= 2 or 3) which cannot be determined by theAssmus-Mattson Theorem; this is the first time that the result by Tang et al. was successfully used to prove that linear codes holdt-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using theCSS construction. As a main method, we use themodified shortening method(simply, calledshortening 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 supportt-designs (t= 2, 3).We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.

It is important to study constructions of plateaued functions with balancedness and high algebraic degrees for preventing cryptographic attacks. Our goal of this paper is to find practical construction methods for producing infinite families of balanced$r$-plateaued functions with maximal algebraic degrees for every positive integer$r$. We first present a theoretical framework for secondary constructions of plateaued functions. From this framework, we derive three practical algorithms by controlling initial input vectors. These algorithms produce$(s+1)$-plateaued functions from a given bent function and$s$-plateaued functions in a recursive way for any nonnegative integer$s$; therefore, we obtain$r$-plateaued functions for every$r > s$. Then we obtain three concrete construction methods of balanced$r$-plateaued functions with maximal algebraic degrees from the algorithms. For implementation, in the tables, we list up some initial bent (0-plateaued) functions, which guarantee the maximality of algebraic degrees of plateaued functions. Furthermore, we discuss the complexities of the three algorithms, which shows the feasibility of our methods. We emphasize that this is the first time to give constructions of balanced$r$-plateaued functions with maximal algebraic degrees for every positive integer$r$as far as we know.