We consider a communication problem in which an update of the source message needs to be conveyed to one or more distant receivers that are interested in maintaining specific linear functions of the source message. The setting is one in which the updates are sparse in nature, and where neither the source nor the receiver(s) is aware of the exact difference vector, but only know the amount of sparsity that is present in the difference-vector. Under this setting, we are interested in devising linear encoding and decoding schemes that minimize the communication cost involved. We show that the optimal solution to this problem is closely related to the notion of maximally recoverable codes (MRCs), which were originally introduced in the context of coding for storage systems. In the context of storage, MRCs guarantee optimal erasure protection when the system is partially constrained to have local parity relations among the storage nodes. In our problem, we show that optimal solutions exist if and only if MRCs of certain kind (identified by the desired linear functions) exist. We consider point-to-point and broadcast versions of the problem, and identify connections to MRCs under both these settings.
Maximally Recoverable Codes: The notion of maximal recoverability in linear codes was originally introduced in @cite_12 . Low field size constructions of partial MDS codes (another name for MRCs with locality) for specific parameter sets codes appear in @cite_2 , @cite_22 , @cite_28 . In the terminology of partial MDS codes, these three works respectively provide low field size constructions for the settings up to one, two and three global parities, and any number of local parities. Explicit constructions based on linearized polynomials for the case of one local parity and any number of global parities appear in @cite_1 . Identifying low-field-size constructions of partial MDS codes for general parameter sets remain an open problem. Various approximations of partial MDS codes like Sector Disk codes @cite_2 , @cite_22 , @cite_28 , STAIR codes @cite_18 , and partial-maximally-recoverable-codes @cite_31 have been considered in literature, and these permit low field size constructions for larger parameter sets. Other known results on maximally recoverable codes include expressions for the weight enumerators of MRCs with locality @cite_13 , and the generalized Hamming weights (GHWs) of the MRSC @math in terms of the GHWs of the code @math @cite_29 .
@cite_18 @cite_22 @cite_28 @cite_29 @cite_1 @cite_2 @cite_31 @cite_13 @cite_12
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An [n, k] linear code C that is subject to locality constraints imposed by a parity check matrix H 0 is said to be a maximally recoverable (MR) code if it can recover from any erasure pattern that some k-dimensional subcode of the null space of H 0 can recover from. The focus in this paper is on MR codes constrained to have all-symbol locality r. Given that it is challenging to construct MR codes having small field size, we present results in two directions. In the first, we relax the MR constraint and require only that apart from the requirement of being an optimum all-symbol locality code, the code must yield an MDS code when punctured in a single, specific pattern which ensures that each local code is punctured in precisely one coordinate and that no two local codes share the same punctured coordinate. We term these codes as partially maximally recoverable (PMR) codes. We provide a simple construction for high-rate PMR codes and then provide a general, promising approach that needs further investigation. In the second direction, we present three constructions of MR codes with improved parameters, primarily the size of the finite field employed in the construction. A code is said to be data-local maximally recoverable if (i) all the information symbols have locality and (ii) any erasure pattern which can be potentially recovered (i.e., the number of equations is equal to the number of unknowns) is recovered by the code. A code is said to be local maximally recoverable if (i) all the symbols of the code have locality and (ii) from the above holds. In this paper, we establish the matroid structures corresponding to data-local and local maximally recoverable codes (MRC). The matroid structures of these codes can be used to determine the associated Tutte polynomial. Greene proved that the weight enumerators of any code can be determined from its associated Tutte polynomial. We will use this result to derive explicit expressions for the weight enumerators of data-local MRC. Also, Britz proved that the higher support weights of any code can be determined from its associated Tutte polynomial. We will use this result to derive expressions for the higher support weights of data-local and local MRC with two local codes. In this paper, we study the maximally recoverable (MR) property for multi-protection group (MPG) codes. MPG codes with MR property achieve the best erasure recoverability given configurations, where a configuration represents the structural relationship between data and parity symbols. We present construction and decoding algorithms for MPG codes with MR property. We show that both recoverability and minimum decoding overhead of any MPG code with MR property depend only on the configuration, where decoding overhead is defined as the additional number of symbols to access, in order to decode the lost data symbols. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes ( , 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math , where @math . The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math . Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property.