Error correction based on partial information

We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded $k$ data symbols using an $(n,k)$ code with code length $n$ and dimension $k.$ During storage, some of the codeword coordinates might be corrupted by errors. We aim to recover the original data by reading the corrupted codeword with a limit on the transmitting bandwidth, namely, we can only download an $\alpha$ proportion of the corrupted codeword. For a given $\alpha,$ our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors. A naive scheme is to read $\alpha n$ coordinates of the codeword. This method used in conjunction with MDS codes guarantees recovery from any $\lfloor(\alpha n-k)/2\rfloor$ errors. In this paper we show that we can instead read an $\alpha$ proportion from each of the codeword's coordinates. For a well-designed MDS code, this method can guarantee recovery from $\lfloor (n-k/\alpha)/2 \rfloor$ errors, which is $1/\alpha$ times more than the naive method, and is also the maximum number of errors that an $(n,k)$ code can correct by downloading only an $\alpha$ proportion of the codeword. We present two families of such optimal constructions and decoding schemes. One is a Reed-Solomon code with evaluation points in a subfield and the other is based on Folded Reed-Solomon codes. We further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword. We also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an $\alpha$ proportion of the corrupted codeword.Comment: Extended version of the conference paper in ISIT 201

Asymptotically MDS Array BP-XOR Codes

Belief propagation or message passing on binary erasure channels (BEC) is a low complexity decoding algorithm that allows the recovery of message symbols based on bipartite graph prunning process. Recently, array XOR codes have attracted attention for storage systems due to their burst error recovery performance and easy arithmetic based on Exclusive OR (XOR)-only logic operations. Array BP-XOR codes are a subclass of array XOR codes that can be decoded using BP under BEC. Requiring the capability of BP-decodability in addition to Maximum Distance Separability (MDS) constraint on the code construction process is observed to put an upper bound on the maximum achievable code block length, which leads to the code construction process to become a harder problem. In this study, we introduce asymptotically MDS array BP-XOR codes that are alternative to exact MDS array BP-XOR codes to pave the way for easier code constructions while keeping the decoding complexity low with an asymptotically vanishing coding overhead. We finally provide and analyze a simple code construction method that is based on discrete geometry to fulfill the requirements of the class of asymptotically MDS array BP-XOR codes.Comment: 8 pages, 4 figures, to be submitte

Asymptotically MDS Array BP-XOR Codes

Belief propagation or message passing on binary erasure channels (BEC) is a low complexity decoding algorithm that allows the recovery of message symbols based on bipartite graph prunning process. Recently, array XOR codes have attracted attention for storage systems due to their burst error recovery performance and easy arithmetic based on Exclusive OR (XOR)-only logic operations. Array BP-XOR codes are a subclass of array XOR codes that can be decoded using BP under BEC. Requiring the capability of BP-decodability in addition to Maximum Distance Separability (MDS) constraint on the code construction process is observed to put an upper bound on the maximum achievable code block length, which leads to the code construction process to become a harder problem. In this study, we introduce asymptotically MDS array BP-XOR codes that are alternative to exact MDS array BP-XOR codes to pave the way for easier code constructions while keeping the decoding complexity low with an asymptotically vanishing coding overhead. We finally provide and analyze a simple code construction method that is based on discrete geometry to fulfill the requirements of the class of asymptotically MDS array BP-XOR codes.Comment: 8 pages, 4 figures, to be submitte