Abstract
We present upper bounds on the size of codes that are locally testable by querying only two input symbols. For linear codes, we show that any 2-locally testable code with minimal distance δn over any finite field \(\mathbb{F}\) cannot have more than \(|\mathbb{F}|^{3/\delta}\) codewords. This result holds even for testers with two-sided error. For general (non-linear) codes we obtain the exact same bounds on the code size as a function of the minimal distance, but our bounds apply only for binary alphabets and one-sided error testers (i.e. with perfect completeness). Our bounds are obtained by examining the graph induced by the set of possible pairs of queries made by a codeword tester on a given code. We also demonstrate the tightness of our upper bounds and the essential role of certain parameters.
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© 2003 Springer-Verlag Berlin Heidelberg
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Ben-Sasson, E., Goldreich, O., Sudan, M. (2003). Bounds on 2-Query Codeword Testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds) Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques. RANDOM APPROX 2003 2003. Lecture Notes in Computer Science, vol 2764. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45198-3_19
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DOI: https://doi.org/10.1007/978-3-540-45198-3_19
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