Variable redundancy product coders

Variable redundancy error detection code

A characterization of MDS codes that have an error correcting pair

Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were found independently by R. K\"otter (1992), as a general algebraic method of decoding linear codes. These pairs exist for several classes of codes. However little or no study has been made for characterizing those codes. This article is an attempt to fill the vacuum left by the literature concerning this subject. Since every linear code is contained in an MDS code of the same minimum distance over some finite field extension we have focused our study on the class of MDS codes. Our main result states that an MDS code of minimum distance $2t+1$ has a $t$-ECP if and only if it is a generalized Reed-Solomon code. A second proof is given using recent results Mirandola and Z\'emor (2015) on the Schur product of codes

Block synchronization for quantum information

Locating the boundaries of consecutive blocks of quantum information is a fundamental building block for advanced quantum computation and quantum communication systems. We develop a coding theoretic method for properly locating boundaries of quantum information without relying on external synchronization when block synchronization is lost. The method also protects qubits from decoherence in a manner similar to conventional quantum error-correcting codes, seamlessly achieving synchronization recovery and error correction. A family of quantum codes that are simultaneously synchronizable and error-correcting is given through this approach.Comment: 7 pages, no figures, final accepted version for publication in Physical Review

Efficiently decoding Reed-Muller codes from random errors

Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length $n=2^m$) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree $r=o(\sqrt{m})$) we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt{m/\log m})$), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$, our algorithm can correct any error pattern in $RM(m,m-(2r+2))$ for which the same erasure pattern can be corrected in $RM(m,m-(r+1))$. The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.Comment: 18 pages, 2 figure

Fast Erasure-and-Error Decoding and Systematic Encoding of a Class of Affine Variety Codes

In this paper, a lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon codes and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Grobner basis, are canonically isomorphic, and moreover, the isomorphism is given by the extension through linear feedback shift registers from Grobner basis and discrete Fourier transforms. Next, the lemma is applied to fast unified system of encoding and decoding erasures and errors in a certain class of affine variety codes.Comment: 6 pages, 2 columns, presented at The 34th Symposium on Information Theory and Its Applications (SITA2011

Passive network tomography for erroneous networks: A network coding approach

Passive network tomography uses end-to-end observations of network communication to characterize the network, for instance to estimate the network topology and to localize random or adversarial glitches. Under the setting of linear network coding this work provides a comprehensive study of passive network tomography in the presence of network (random or adversarial) glitches. To be concrete, this work is developed along two directions: 1. Tomographic upper and lower bounds (i.e., the most adverse conditions in each problem setting under which network tomography is possible, and corresponding schemes (computationally efficient, if possible) that achieve this performance) are presented for random linear network coding (RLNC). We consider RLNC designed with common randomness, i.e., the receiver knows the random code-books all nodes. (To justify this, we show an upper bound for the problem of topology estimation in networks using RLNC without common randomness.) In this setting we present the first set of algorithms that characterize the network topology exactly. Our algorithm for topology estimation with random network errors has time complexity that is polynomial in network parameters. For the problem of network error localization given the topology information, we present the first computationally tractable algorithm to localize random errors, and prove it is computationally intractable to localize adversarial errors. 2. New network coding schemes are designed that improve the tomographic performance of RLNC while maintaining the desirable low-complexity, throughput-optimal, distributed linear network coding properties of RLNC. In particular, we design network codes based on Reed-Solomon codes so that a maximal number of adversarial errors can be localized in a computationally efficient manner even without the information of network topology.Comment: 40 pages, under submission for IEEE Trans. on Information Theor