In this paper, we introduce a novel explicit family of subcodes of Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a constant output list size. Our approach builds upon the idea of large linear subcodes of RS codes evaluated on a subfield, similar to the method employed by Guruswami and Xing (STOC 2013). However, our approach diverges by leveraging the idea of {\it permuted product codes}, thereby simplifying the construction by avoiding the need of {\it subspace designs}. Specifically, the codes are constructed by initially forming the tensor product of two RS codes with carefully selected evaluation sets, followed by specific cyclic shifts to the codeword rows. This process results in each codeword column being treated as an individual coordinate, reminiscent of prior capacity-achieving codes, such as folded RS codes and univariate multiplicity codes. This construction is easily shown to be a subcode of an interleaved RS code, equivalently, an RS code evaluated on a subfield. Alternatively, the codes can be constructed by the evaluation of bivariate polynomials over orbits generated by \emph{two} affine transformations with coprime orders, extending the earlier use of a single affine transformation in folded RS codes and the recent affine folded RS codes introduced by Bhandari {\it et al.} (IEEE T-IT, Feb.~2024). While our codes require large, yet constant characteristic, the two affine transformations facilitate achieving code length equal to the field size, without the restriction of the field being prime, contrasting with univariate multiplicity codes.

翻译：本文提出了一类新的Reed-Solomon(RS)码显式子码族，能以恒定输出列表大小高效实现列表译码容量。我们的方法建立在RS码在子域上求值的大规模线性子码思想之上，类似于Guruswami和Xing（STOC 2013）所采用的方法。然而，我们的方法通过利用"置换乘积码"思想实现了突破，通过避免使用"子空间设计"简化了构造。具体而言，码的构造过程首先将两个具有精心选择求值集的RS码进行张量积运算，然后对码字行施加特定的循环移位。此过程使得每个码字列被视作独立坐标，类似于先前的容量可达码（如折叠RS码和单变量重数码）。该构造易于证明为交织RS码的子码，等价于在子域上求值的RS码。另一种构造方式是通过对由两个互质阶仿射变换生成的轨道上的二元多项式求值实现，这扩展了折叠RS码中单一仿射变换的先前应用以及Bhandari等人（IEEE T-IT, 2024年2月）近期提出的仿射折叠RS码。虽然我们的码需要较大但恒定的特征值，但与单变量重数码不同，这两个仿射变换使得码长可达域大小，且不受域为素数域的限制。