Linear Exact Repair in MDS Array Codes: A General Lower Bound and Its Attainability

Abstract

For an $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$, how small can the repair bandwidth and repair I/O be under linear exact repair? We study this question in the regime where the field size $q$, the redundancy $r=n-k$, and the sub-packetization level $\ell$ are fixed, while the code length $n$ varies, and we develop a geometric approach to this setting. Our starting point is an intrinsic reformulation of linear exact repair for MDS array codes in terms of subspace intersections and, for repair I/O, the projective point configurations induced by a parity-check realization. This viewpoint yields a simple projective counting argument establishing the general lower bound $$\beta_{\mathrm{avg}},\beta_{\max},\gamma_{\mathrm{avg}},\gamma_{\max}\;\ge\;\ell(n-1)-\frac{q^{(r-1)\ell}-1}{q-1}$$ for linear exact repair of every $(n,k,\ell)$ MDS array code over $\mathbb{F}_q$ with redundancy $r=n-k\ge 2$. To our knowledge, this is the first lower bound of this form that applies to arbitrary redundancy $r\ge 2$ and sub-packetization level $\ell$. At first glance, the projective counting bound appears rather coarse and therefore unlikely to be attained. We prove that this intuition is correct whenever $r\ge 3$ and $\ell\ge 2$. For $r=2$, the picture changes completely. Using Desarguesian spreads from finite geometry, we construct MDS array codes that attain the bound over a broad interval of code lengths, up to the maximum possible length $q^{\ell}+1$, and do so simultaneously for both repair bandwidth and repair I/O. In the smallest nontrivial case $(r,\ell)=(2,2)$, we also prove a converse within the regular-spread model. Together, these results identify a uniform obstruction governing linear exact repair and show that, in the two-parity case, this obstruction is tight.