This paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality. Keywords Hierarchical locality · Alphabet-dependent bound · Simplex code · Matroid theory Mathematics Subject Classification 94B65 · 94B60 · 05B35 Introduction In modern distributed

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... orage systems (DSSs) failures happen frequently, whence decreasing the number of connections required for node repair is crucial. Locally repairable codes (LRCs) are a subclass of erasure-correcting codes, which allow a small number of failed nodes to be repaired by accessing only a few other nodes. LRCs were introduced in [6], [11] where the codes can locally repair one failure. They were later extended in [12], [8] to be able to repair more failures locally. An [n, k, d] linear code C of length n, dimension k, and minimum Hamming distance d, has all-symbol locality (r , δ) if for all code symbols i ∈ [n] = {1, . . . , n}, there exists a set R i ⊆ [n] containing i such that |R i | ≤ r + δ − 1 and the minimum distance of the restriction of C to R i is at least δ. We refer to C as an (n, k, d, r , δ)-LRC and to the sets R i as Communicated by T. Etzion. Part of the results were submitted without any proofs to IEEE International Symposium Information Theory (ISIT) 2019.