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Revision #3 to TR11-087 | 10th June 2012 19:09

#### A Combination of Testability and Decodability by Tensor Products

Revision #3
Authors: Michael Viderman
Accepted on: 10th June 2012 19:09
Keywords:

Abstract:

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as a black box) to present a combinatorial construction of locally testable codes that match best known parameters. As a consequence, this construction was obtained over sufficiently large fields.

In this paper we improve the result of Ben-Sasson and Sudan and show that for \emph{any} linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field.

Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties:
\begin{itemize}
\item have constant rate and constant relative distance;
\item have blocklength $n$ and testable with $n^{\epsilon}$ queries, for any constant $\epsilon > 0$;
\item linear time encodable and linear-time decodable from a constant fraction of errors.
\end{itemize}

Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary regarding the list-decodable codes.

Revision #2 to TR11-087 | 6th November 2011 18:51

#### A Combination of Testability and Decodability by Tensor Products

Revision #2
Authors: Michael Viderman
Accepted on: 6th November 2011 18:51
Keywords:

Abstract:

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as a black box) to present a combinatorial construction of locally testable codes that match best known parameters. As a consequence, this construction was obtained over sufficiently large fields.

In this paper we improve the result of Ben-Sasson and Sudan and show that for \emph{any} linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field.

Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties:
\begin{itemize}
\item have constant rate and constant relative distance;
\item have blocklength $n$ and testable with $n^{\epsilon}$ queries, for any constant $\epsilon > 0$;
\item linear time encodable and linear-time decodable from a constant fraction of errors.
\end{itemize}

Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary regarding the list-decodable codes.

Revision #1 to TR11-087 | 3rd August 2011 19:31

#### A Combination of Testability and Decodability by Tensor Products

Revision #1
Authors: Michael Viderman
Accepted on: 3rd August 2011 19:31
Keywords:

Abstract:

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be
applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as a black box) to present a combinatorial construction of locally testable codes that match best known parameters. As a consequence,
this construction was obtained over sufficiently large fields.

In this paper we improve the result of Ben-Sasson and Sudan and show that for \emph{any} linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field.

Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties:
\begin{itemize}
\item have constant rate and constant relative distance;
\item have blocklength $n$ and testable with $n^{\epsilon}$ queries, for any constant $\epsilon > 0$;
\item linear time encodable and linear-time decodable from a constant fraction of errors.
\end{itemize}

Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary regarding the list-decodable codes.

### Paper:

TR11-087 | 3rd June 2011 21:52

#### A Combination of Testability and Decodability by Tensor Products

TR11-087
Authors: Michael Viderman
Publication: 3rd June 2011 22:41
Keywords:

Abstract:

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as a black box) to present a combinatorial construction of locally testable codes that match best known parameters. As a consequence, this construction was obtained over sufficiently large fields.

In this paper we improve the result of Ben-Sasson and Sudan and show that for \emph{any} linear codes the associated tensor products are locally testable. Consequently, the construction of Meir can be taken over any field, including the binary field.

Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a construction of linear codes (over any field) that combine the following properties:
\begin{itemize}
\item have constant rate and constant relative distance;
\item have blocklength $n$ and testable with $n^{\epsilon}$ queries, for any constant $\epsilon > 0$;
\item linear time encodable and linear-time decodable from a constant fraction of errors.
\end{itemize}

Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies a similar corollary regarding the list-decodable codes.

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