Bounds on the minimum distance of locally recoverable codes

TL;DR

The paper addresses the problem of determining the minimal length $n_q(k,d,r)$ of linear $[n,k,d]_q$-codes with locality $r$, situating it within classical bounds and new geometric insights. It introduces a geometric reformulation in $\mathrm{PG}(k-1,q)$, showing that for fixed $q,k,r$ the minimal length equals the Griesmer bound $g_q(k,d)$ once $d$ is sufficiently large, and provides ILP-based methods to compute exact values for small parameters. It delivers exact small-parameter results over binary and partial data for ternary fields, along with broad constructions (simplex, $t$-fold simplex, Solomon–Stiffler) that attain or bound these values, and it uses enumeration to empirically map the locality landscape. The work thus connects projective geometry with LRC design, offering exact benchmarks, scalable constructions, and practical guidance for storage-related applications where small locality with large distance is desirable.

Abstract

We consider locally recoverable codes (LRCs) and aim to determine the smallest possible length $n=n_q(k,d,r)$ of a linear $[n,k,d]_q$-code with locality $r$. For $k\le 7$ we exactly determine all values of $n_2(k,d,2)$ and for $k\le 6$ we exactly determine all values of $n_2(k,d,1)$. For the ternary field we also state a few numerical results. As a general result we prove that $n_q(k,d,r)$ equals the Griesmer bound if the minimum Hamming distance $d$ is sufficiently large and all other parameters are fixed.

Theorems & Definitions (67)

• Definition 1
• Lemma 2
• Lemma 3
• Definition 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• proof
• Example 9