Exponents of polar codes using algebraic geometric code kernels

Reed-Solomon and BCH codes were considered as kernels of polar codes by Mori and Tanaka in [6] and Korada, Şaşoğlu, and Urbanke in [4] to create polar codes with large exponents. For codes of length l ≤ q, Mori and Tanaka showed Reed-Solomon codes give the best possible exponent, where q is the size of the input alphabet. However, considering larger matrices over Fq yields better exponents for longer codes. We expand on these ideas by using AG codes as kernels of polar codes. We provide a lower bound on the exponent of such kernels. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than comparable Reed-Solomon codes, for q > 4.