Doctoral defence: Junming Ke "Codes for Distributed Storage"

Summary 
Vast amounts of data are generated from sources like video, voice, and text, and these amounts are still growing yearly, requiring sustainable storage systems for data integrity and availability. Distributed Storage Systems (DSS's) offer a low-cost, reliable, and stable solution. A DSS stores data across potentially unreliable storage nodes by adding redundancy to ensure system reliability. Efficient update and repair mechanisms are critical for maintaining stability, especially during node failures. This thesis first constructs update-efficient codes based on finite projective planes, featuring efficient local repair and large repair availability. Sparse generator and parity-check matrices are provided that also allow efficient updates with flexibility in update frequencies. The performance of a DSS using these codes is also evaluated. Two key concepts in storage codes are node capacity and repair bandwidth, where the possible trade-offs between them are determined by the cutset bound. Most research focuses on exact repair, where lost node data is recovered exactly. However, functional repair, which does not recover lost node content exactly but still maintains data integrity, allows to reach otherwise unattainable points on the cutset bound. This thesis constructs a new optimal functional-repair code based on a vector space partition in 9-dimensional binary space, strongly related to the projective plane $\PG(2,8)$. This storage code also has an efficient repair algorithm which is described both geometrically and algebraically.