Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
Pseudorandom generators without the XOR lemma
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Semi-Direct Product in Groups and Zig-Zag Product in Graphs: Connections and Applications
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
A new family of Cayley expanders (?)
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Combinatorica
Simple PCPs with poly-log rate and query complexity
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Some 3CNF Properties Are Hard to Test
SIAM Journal on Computing
Almost Orthogonal Linear Codes are Locally Testable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Locally testable codes and PCPs of almost-linear length
Journal of the ACM (JACM)
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Combinatorial construction of locally testable codes
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Algebraic property testing: the role of invariance
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
IEEE Transactions on Information Theory - Part 1
Improved low-density parity-check codes using irregular graphs
IEEE Transactions on Information Theory
The capacity of low-density parity-check codes under message-passing decoding
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides new way of constructions with better parameters and new lower bounds. Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates "look the same", namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all "low-degree" codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a "conflict" between 1/rate and density (constraint length). In known constructions, if one is constant then the other is almost worst possible - n/poly(log n). Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log log n)2, of constant rate and with constraints of length (log n)4. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes. Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka "good" codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by log(ω(l)) n if the group has a derived series of length l This negative result precludes natural local tests with constantly many queries for such solvable "good" codes.