极化解减单项式-笛卡尔码
Polar decreasing monomial-Cartesian codes
AI总结：
本文证明多核极化码可统一视为极化解减单项式-笛卡尔码，扩展极化定理至任意域对称信道，并给出对偶性质及码参数描述。
摘要：
我们证明，在特定对称信道上具有多个核的极化码族可以被视为极化解减单项式-笛卡尔码，从而为任意有限域上的此类码提供统一处理。我们将减单项式-笛卡尔码定义为在笛卡尔积上对一组在整除关系下封闭的单项式求值。极化解减单项式-笛卡尔码是减单项式-笛卡尔码，其单项式集合关于由Bardet、Dragoi、Otmani和Tillich近期工作["Algebraic properties of polar codes from a new polynomial formalism," 2016 IEEE International Symposium on Information Theory (ISIT)]启发的偏序封闭。扩展Mori和Tanaka的主要定理["Source and Channel Polarization Over Finite Fields and Reed-Solomon Matrices," in IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2720--2736, May 2014]，我们证明任意域上满足特定条件的任何可逆矩阵序列都能使该域上任何对称信道极化。此外，我们证明减单项式-笛卡尔码的对偶码在单项式上等价于一个减单项式-笛卡尔码。定义单项式集合的最小生成集，我们利用它来描述减单项式-笛卡尔码的长度、维数和最小距离。
英文摘要：
We prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes, offering a unified treatment for such codes, over any finite field. We define decreasing monomial-Cartesian codes as the evaluation of a set of monomials closed under divisibility over a Cartesian product. Polar decreasing monomial-Cartesian codes are decreasing monomial-Cartesian codes whose sets of monomials are closed respect a partial order inspired by the recent work of Bardet, Dragoi, Otmani, and Tillich ["Algebraic properties of polar codes from a new polynomial formalism," 2016 IEEE International Symposium on Information Theory (ISIT)]. Extending the main theorem of Mori and Tanaka ["Source and Channel Polarization Over Finite Fields and Reed-Solomon Matrices," in IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2720--2736, May 2014], we prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any symmetric over the field channel. In addition, we prove that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Defining the minimal generating set for a set of monomials, we use it to describe the length, dimension and minimum distance of a decreasing monomial-Cartesian code.
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