We discover a novel connection between two classical mathematical notions, Eulerian orientations and Hadamard codes by studying the counting problem of Eulerian orientations (\#EO) with local constraint functions imposed on vertices. We present two special classes of constraint functions and a chain reaction algorithm, and show that the \#EO problem defined by each class alone is polynomial-time solvable by the algorithm. These tractable classes of functions are defined inductively, and quite remarkably the base level of these classes is characterized perfectly by the well-known Hadamard code. Thus, we establish a novel connection between counting Eulerian orientations and coding theory. We also prove a \#P-hardness result for the \#EO problem when constraint functions from the two tractable classes appear together.

翻译：通过研究在顶点上施加局部约束函数的欧拉定向计数问题（#EO），我们发现了两个经典数学概念——欧拉定向与哈达玛码——之间的新颖联系。我们提出了两类特殊的约束函数及一种链式反应算法，并证明由任一类别单独定义的#EO问题均可通过该算法在多项式时间内求解。这些易处理的函数类别采用归纳方式定义，而令人瞩目的是，这些类别的基层次恰好由著名的哈达玛码完美表征。由此，我们在欧拉定向计数与编码理论之间建立了全新联系。同时，当两类易处理约束函数同时出现时，我们证明了#EO问题的#P难解性。