From Linear Exact Repair to Mixed-Alphabet Codes: The Bandwidth-Subpacketization Tradeoff of MDS Array Codes

Abstract

MDS array codes achieve optimal storage efficiency in distributed storage systems, but their repair bandwidth depends critically on the sub-packetization level. While the theoretically optimal repair bandwidth (i.e., cut-set bound) of a high-rate exact-repair MDS array code is attainable with sufficiently large sub-packetization, the fundamental limits for fixed sub-packetization remain poorly understood. We study this problem for centralized linear exact repair of $h$ failures from $d$ helper nodes over a fixed finite field. We show that the rank profile of any linear repair scheme is constrained by two auxiliary mixed-alphabet codes. The first code is formed from the kernels of the helper projections and exists for all repair degrees; the second is formed from their row spaces and applies in the full-helper regime when $d\ge r$. These codes have complementary alphabet profiles, and their size and minimum-distance requirements are forced solely by the MDS property and repair correctness. This gives a unified route from classical coding-theoretic upper bounds to repair-bandwidth lower bounds. As a consequence, we obtain several explicit lower bounds via Singleton, Hamming, Plotkin, and Griesmer arguments, including known bounds as corollaries. We further formulate a Delsarte-type support-enumerator LP for mixed-alphabet codes and give an exact type-compressed implementation. For fixed sub-packetization \(\ell\), this yields a polynomial-time procedure for evaluating the resulting finite-parameter lower bound.