A framework of monomial codes is considered, which includes linear codes generated by the evaluation of certain monomials. Polar and Reed-Muller codes are the two best-known representatives of such codes and can be considered as two extreme cases. Reed-Muller codes have a large automorphism group but their low-complexity maximum likelihood decoding still remains an open problem. On the other hand, polar codes have much less symmetries but admit the efficient near-ML decoding. We study the dependency between the code symmetries and the decoding efficiency. We introduce a new family of codes, partially symmetric monomial codes. These codes have a smaller group of symmetries than the Reed-Muller codes and are in this sense "between" RM and polar codes. A lower bound on their parameters is introduced along with the explicit construction which achieves it. Structural properties of these codes are demonstrated and it is shown that they often have a recursive structure.
翻译：单一代码框架被考虑,它包括对某些单项代码的评估所产生的线性代码。极代码和Reed-Muller代码是这类代码的两个最著名的代表,可被视为两个极端情况。Reed-Muller代码有一个很大的自制组,但其低复杂度最大解码可能性仍然是一个尚未解决的问题。另一方面,极地代码的对称性要小得多,但承认了近ML解码的效率。我们研究了代码对称性和解码效率之间的依赖性。我们引入了一套新的代码,部分对称单项代码。这些代码比Reed-Muler代码的对称性要小,从这个意义上说,它们属于“RM和极地代码之间”的对称性,因此,它们的参数与实现这些代码的清晰构造相连接要小一些。这些代码的结构特性得到了证明,并且表明它们往往具有连成结构。
相关内容
微信扫码咨询专知VIP会员