论文标题
极性降低单一 - 牙犯代码
Polar decreasing monomial-Cartesian codes
论文作者
论文摘要
我们证明，在某些对称通道上具有多个内核的极性代码家族可以看作是极性减少单一式 - 卡ac式代码，并在任何有限领域都提供了针对此类代码的统一处理。我们将降低的单一 - 皇家法规定义为对一组在笛卡尔产品上可划分的单一封闭性的评估。极性降低单一式 - 哥伦比亚式代码正在减少单一级别的法规，其封闭式的封闭式尊重部分秩序，这是受Bardet，Dragoi，Otmani和Tillich的最新作品的启发，[“ 2016 IEE EEE International Sympos Inspectim of Insvertion Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Inspection of Insports of Insvertion（Ispos）。在信息理论的IEEE交易中，扩展了Mori和Tanaka的主要定理[“在有限场上和芦苇 - 固体矩阵上的源和通道极化”，第1卷。 60，没有。 5，第2720--2736页，2014年5月]，我们证明，在某些条件下满足某些条件的任意磁场上的任何可逆矩阵序列都在野外通道上偏振。此外，我们证明了降低的单一 - 帕特斯密码的双重二重性在单一上等同于减少的单体 - 卡丁裔代码。为了定义一组单元的最小生成集，我们使用它来描述减少单元 - 卡丁裔代码的长度，尺寸和最小距离。
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.
