Abstract
A k-plex of order n is an n × n matrix on n symbols, where every row contains k distinct symbols, every column contains k distinct symbols, and every symbol occurs exactly k times. Yi et al. (2019) introduced 3-plex codes which are 2-erasure codes (2-erasure tolerant array codes) derived from 3-plexes. In this paper, we generalize 3-plex codes to k-plex codes. We introduce the notion of a “strong” k-plex which implies the derived k-plex code is 2-erasure tolerant. Moreover, k-plex codes derived from strong k-plexes have a straightforward algorithm for reconstruction. These general k-plex codes offer greater flexibility when choosing a suitable code for a storage system, enabling the operator to better optimize the unavoidable trade-offs involved. Blackburn asked for the maximum number of entries in an n × n partial Latin square on n symbols in which if distinct cells (i, j) and (i', j') contain the same symbol, then the cells (i', j) and (i, j') are empty. A “strong” k-plex satisfies the Blackburn property (along with two other properties related to erasure coding). We investigate the necessary conditions for the existence of Blackburn k-plexes (and hence necessary conditions for the existence of strong k-plexes). We show that any Blackburn k-plex has order n ≥ [(√2 + 1)k - 2]. We describe how to construct strong k-plexes of order n when k ∈ {2, 3, 4, 5} for all possible orders n, and we give a simple construction of strong k-plexes of order k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for k ≥ 2.
Highlights
A RRAY codes are a type of erasure code whose parity blocks are determined by a matrix
For k-plex codes with k ≥ 4, we reduce the storage overhead, but we increase the reconstruction cost; practicality depends on the size of the storage system and the frequency of erasures
Blackburn partial Latin squares are related to parent-identifying codes [1] and perfect hash families of strength 3 with three rows [2], [9], [14], [16]
Summary
A RRAY codes are a type of erasure code whose parity blocks are determined by a matrix. A partial Latin square is an n × n matrix with symbols from Zn and possibly empty cells, in which each row contains distinct symbols, and each column contains distinct symbols. We consistently index the rows and columns by Zn, and take the symbol set as Zn. For studying the arising k-plex codes, it is important that these three indexing sets are all equal. Wanless [22] described the Blackburn property for partial Latin squares: if distinct cells (i, j) and (i, j) contain the same symbol, cells (i, j) and (i, j) are both empty. We prove mathematical properties relating to the erasure tolerance of k-plex codes in general, rather than in a particular practical instance as in [26].
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