基于Hadamard矩阵的最优局部修复码构造

现有的局部修复码大多能满足最小距离最优的边界条件，但是在满足最小距离最优情况下构造维度最优的局部修复码还比较困难。针对上述问题，提出一种基于Hadamard矩阵的最优局部修复码的构造方法，通过对Hadamard矩阵进行扩展，构造局部修复码的校验矩阵，进而通过此校验矩阵构造最优局部修复码。首先，基于Hadamard矩阵构造局部修复码的校验矩阵，通过校验矩阵构造的局部修复码的最小距离可以达到最优最小距离界，但是其维度没有达到最优维度边界条件；为进一步提高维度，将校验矩阵中的关联矩阵0和1元素互换得到新的关联矩阵，通过和新的关联矩阵级联进行扩展，构造的扩展局部修复码不仅可以达到最小距离最优，且能达到维度最优的边界条件。与现有局部修复码相比，该构造的局部修复码是最小距离和维度最优的局部修复码，且其码率也更逼近局部修复码最优码率的边界。

Construction of Optimal Locally Repairable Codes Based on Hadamard Matrix

Most of the existing locally repairable codes can meet the boundary condition of minimum distance, but it is difficult to construct the locally repairable codes with optimal dimension under the condition of minimum distance optimization. To solve this problems, this paper proposes a construction method of optimal locally repairable codes based on Hadamard matrix. By expanding Hadamard matrix, the check matrix of the optimal locally repairable code can be obtained. Specifically, the parity matrix of locally repairable codes is constructed using Hadamard matrix, and the minimum distance of the locally repairable codes constructed by the parity matrix can reach the optimal boundary, but its dimension does not reach the optimal dimension boundary condition. In order to improve the dimension, the element 0 and element 1 of the incidence matrix in the check matrix are exchanged to obtain a new incidence matrix. By cascading with the new incidence matrix, the constructed extended locally repairable code can not only achieve the minimum distance optimization, but also achieve the boundary condition of the optimal dimension. Compared with the existing locally repairable codes, the extended locally repairable code based on Hadamard matrix constructed in this paper is the optimal locally repairable code with minimum distance and dimension, and its code rate is closer to the boundary of the optimal code rate of locally repairable codes.