Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices

We prove that for any positive integers $n$ and $k$ such that $n\!\geq\! k\!\geq\! 1$, there exists an $[n,k]$ generalized Reed-Solomon (GRS) code that has a sparsest and balanced generator matrix (SBGM) over any finite field of size $q\!\geq\! n\!+\!\lceil\frac{k(k-1)}{n}\rceil$, where sparsest means that each row of the generator matrix has the least possible number of nonzeros, while balanced means that the number of nonzeros in any two columns differ by at most one. Previous work by Dau et al (ISIT'13) showed that there always exists an MDS code that has an SBGM over any finite field of size $q\geq {n-1\choose k-1}$, and Halbawi et al (ISIT'16, ITW'16) showed that there exists a cyclic Reed-Solomon code (i.e., $n=q-1$) with an SBGM for any prime power $q$. Hence, this work extends both of the previous results

Balanced and sparse Tamo-Barg codes

We construct balanced and sparse generator matrices for Tamo and Barg's Locally Recoverable Codes (LRCs). More specifically, for a cyclic Tamo-Barg code of length n, dimension k and locality r, we show how to deterministically construct a generator matrix where the number of nonzeros in any two columns differs by at most one, and where the weight of every row is d + r − 1, where d is the minimum distance of the code. Since LRCs are designed mainly for distributed storage systems, the results presented in this work provide a computationally balanced and efficient encoding scheme for these codes. The balanced property ensures that the computational effort exerted by any storage node is essentially the same, whilst the sparse property ensures that this effort is minimal. The work presented in this paper extends a similar result previously established for Reed-Solomon (RS) codes, where it is now known that any cyclic RS code possesses a generator matrix that is balanced as described, but is sparsest, meaning that each row has d nonzeros

Error-Correcting Codes for Networks, Storage and Computation

The advent of the information age has bestowed upon us three challenges related to the way we deal with data. Firstly, there is an unprecedented demand for transmitting data at high rates. Secondly, the massive amounts of data being collected from various sources needs to be stored across time. Thirdly, there is a need to process the data collected and perform computations on it in order to extract meaningful information out of it. The interconnected nature of modern systems designed to perform these tasks has unraveled new difficulties when it comes to ensuring their resilience against sources of performance degradation. In the context of network communication and distributed data storage, system-level noise and adversarial errors have to be combated with efficient error correction schemes. In the case of distributed computation, the heterogeneous nature of computing clusters can potentially diminish the speedups promised by parallel algorithms, calling for schemes that mitigate the effect of slow machines and communication delay. This thesis addresses the problem of designing efficient fault tolerance schemes for the three scenarios just described. In the network communication setting, a family of multiple-source multicast networks that employ linear network coding is considered for which capacity-achieving distributed error-correcting codes, based on classical algebraic constructions, are designed. The codes require no coordination between the source nodes and are end to end: except for the source nodes and the destination node, the operation of the network remains unchanged. In the context of data storage, balanced error-correcting codes are constructed so that the encoding effort required is balanced out across the storage nodes. In particular, it is shown that for a fixed row weight, any cyclic Reed-Solomon code possesses a generator matrix in which the number of nonzeros is the same across the columns. In the balanced and sparsest case, where each row of the generator matrix is a minimum distance codeword, the maximal encoding time over the storage nodes is minimized, a property that is appealing in write-intensive settings. Analogous constructions are presented for a locally recoverable code construction due to Tamo and Barg. Lastly, the problem of mitigating stragglers in a distributed computation setup is addressed, where a function of some dataset is computed in parallel. Using Reed-Solomon coding techniques, a scheme is proposed that allows for the recovery of the function under consideration from the minimum number of machines possible. The only assumption made on the function is that it is additively separable, which renders the scheme useful in distributed gradient descent implementations. Furthermore, a theoretical model for the run time of the scheme is presented. When the return time of the machines is modeled probabilistically, the model can be used to optimally pick the scheme's parameters so that the expected computation time is minimized. The recovery is performed using an algorithm that runs in quadratic time and linear space, a notable improvement compared to state-of-the-art schemes. The unifying theme of the three scenarios is the construction of error-correcting codes whose encoding functions adhere to certain constraints. It is shown that in many cases, these constraints can be satisfied by classical constructions. As a result, the schemes presented are deterministic, operate over small finite fields and can be decoded using efficient algorithms.</p