Exponents of polar codes using algebraic geometric code kernels

Abstract

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 F q with q elements give the best possible exponent among all codes of length l ≤ q. They also stated that a Hermitian code over F 2 r with r ≥ 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 ≥ 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.

Vardøhus Codes: Polar Codes Based on Castle Curves Kernels

“…Next theorem was proved by Anderson and Matthews in [1]. It says that shortening kernels from algebraic curves does not change the final structure of the code.…”

“…In Section II we have compile some basic facts on polar codes and algebraic geometric curves needed to understand the paper. Section III reviews some results in [1] and we adapt them for constructing kernel matrices that arise from algebraic curves for a SOF channel that is a discrete memoryless channel which is symmetric w.r.t the field operations. Section IV deals with the computation of the minimum distance and the dual of codes proposed in the previous section.…”

“…, where G i is the i-th row of G. The original Arıkan's matrix G A has exponent 1 2 . The exponent is the value such that…”

“…Anderson and Matthews proved in [1] that this means that for any β < E(G) polar coding using kernel G over a DMC channel W at a fixed rate 0 < R < I(W ) and block length N = l n implies…”

Channel Coding Toward 6G: Technical Overview and Outlook

“…Different linear non-binary kernels have also been proposed. In [468], non-binary kernel matrices were constructed based on Reed-Solomon codes and Hermitian codes that were later expanded to other algebraic geometry codes, including Suzuki codes, in [469], concatenated algebraic geometry codes, due to their large minimum distance and often nested structure, in [470]. These kernels have a larger exponent than any linear binary kernel of the same dimension.…”