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 (IEEE Information Theory Workshop, 2010, pp 1–5) and Korada et al. (IEEE Trans Inform Theory 56(12):6253–6264, 2010) to create polar codes with large exponents. Mori and Tanaka showed that Reed–Solomon codes over the finite field \(\mathbb {F}_q\) with \(q\) elements give the best possible exponent among all codes of length \(l \le q\). They also stated that a Hermitian code over \(\mathbb {F}_{2^r}\) with \(r \ge 4\), a simple algebraic geometric code, gives a larger exponent than the Reed–Solomon matrix over the same field. In this paper, we expand on these ideas by employing more general algebraic geometric (AG) codes to produce kernels of polar codes. Lower bounds on the exponents are given for kernels from general AG codes, Hermitian codes, and Suzuki codes. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than Reed–Solomon codes over the same field, for \(q \ge 3\); however, the larger exponents are at the expense of larger kernel matrices. Comparing kernels of the same size, though over different fields, we see that Reed–Solomon kernels have larger exponents than both Hermitian and Suzuki kernels. These results indicate a tradeoff between the exponent, kernel matrix size, and field size.