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Explicit Constructions of Optimal-Access MDS Codes With Nearly Optimal Sub-Packetization

机译：具有接近最佳子分组的最优访问MDS代码的显式构造

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An maximum distance separable (MDS) array code of length , dimension , and sub-packetization is formed of matrices over a finite field , with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with , the sub-packetization satisfies the bound . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any and , we presented an explicit construction of optimal-access MDS codes with sub-packetization . In this paper, we take up the question of reducing the sub-packetization value to make it to approach the lower bound. We construct an explicit family of optimal-access codes with , which differs from the optimal value by at most a factor of . These codes can be constructed over any finite field as long as , and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access. In this variation, we assume that the set of nodes is partitioned into repair groups of size , and require that the amount of accessed data for repair is the smallest possible whenever the helper nodes include all the other nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field of size , and also afford low-complexity encoding and decoding procedures.
机译：长度，维数和子分组化的最大距离可分离（MDS）数组代码由有限域上的矩阵组成，矩阵的每一列都存储在分布式存储系统中的单独节点上，并视为矩阵的坐标。码字。可以通过访问一组尚存的（辅助）节点来执行故障节点的修复（恢复一个删除的列）。如果在每个辅助节点上访问的数据量满足此数量的下限，则该代码具有最佳访问属性。对于具有的最佳访问MDS代码，子分组满足bound。在我们之前的工作中（IEEE Trans。Inf。Theory，第63卷，第4期，2017年），对于任何一个，我们都提出了带有子分组的最优访问MDS代码的显式构造。在本文中，我们讨论了降低子分组化值使其接近下限的问题。我们使用构造了一个明确的最佳访问代码族，它与最佳值的相差最多为。只要，这些代码就可以在任何有限字段上构造，并提供低复杂度的编码和解码过程。我们还定义了修复问题的一种版本，它将重新生成代码的代码与具有局域性约束的代码（LRC代码）联系起来，我们称之为具有最佳访问权限的组修复。在此变体中，我们假定将节点集合划分为大小为的修复组，并要求每当帮助者节点包括与故障节点属于同一组的所有其他节点时，用于修复的访问数据量应尽可能小。 。对于此问题，我们构造了具有组最佳访问属性的代码族。这些代码可以在任何大小的字段上构造，并且还提供低复杂度的编码和解码过程。

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