Fault-Tolerant Dot-Product Engines

Coding schemes are presented that provide the ability to correct and detect computational errors while using dot-product engines for integer vector--matrix multiplication. Both the $L_1$-metric and the Hamming metric are considered

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was submitted to IEEE Transactions on Information Theory on January 21, 2005. To appear in IEEE Transactions on Information Theory, August 2006. 12 page

Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs

A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood

1 Burst List Decoding of Interleaved Reed–Solomon Codes

Abstract—It is shown that interleaved Reed–Solomon codes can be list-decoded for burst errors while attaining the generalized Reiger bound for list decoding. A respective decoding algorithm is presented which is (significantly) more efficient than a burst list decoder for a non-interleaved Reed–Solomon code with comparable parameters. Finally, it is shown through counterexamples that, unlike the special case of Reed–Solomon codes, interleaving does not always preserve the list decoding properties of the constituent code. Index Terms—Burst errors, interleaving, list decoding, Reed– Solomon codes, Reiger bound. I

1 Bounds and Constructions for Granular Media Coding

Abstract—Bounds on the rates of grain-correcting codes are presented. The lower bounds are Gilbert–Varshamov-like ones, whereas the upper bounds improve on the previously known result by Mazumdar et al. Constructions of t-grain-correcting codes of length n for certain values of n and t are discussed. Finally, an infinite family of codes of rate approaching 1 that can detect an arbitrary number of grain errors is shown to exist. Index Terms—convex optimization, Gilbert–Varshamov bound, grain-correcting codes, granular media, lower bounds, magnetic recording, Markov chain, upper bounds. I