On a Question of Cyclic Codes With Generalized Niho-Type Nonzeroes

We compute the weight distribution of binary cyclic codes with arbitrarily many generalized Niho-type nonzeroes, where the exponents of the nonzeroes nearly form a long arithmetic progression. Determining their weight distribution requires two distinct techniques. The first involves using the polar representation associated with generalized Niho-type exponents and leveraging the subtle structure of the nonzeroes whose exponents are close to an arithmetic progression. The second applies the weight distribution of binary Zetterberg codes to address certain counting problems. This resolves two open questions raised by Li and Zeng.