Symmetric LDPC codes and local testing

Abstract

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction 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 “con ict” between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn).

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 logn)2, of constant rate and with constraints of length (logn)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)(an Ω(ℓ) iterated logarithm) 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.

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References

  1. [1]

    N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn and D. Ron: Testing Low Degree Polynomials Over GF(2), in: Proceedings of 7th International Workshop on Randomization and Computation, (RANDOM), Lecture Notes in Computer Science 2764 (2003), 188-199. Also, IEEE Transactions on Information Theory, 51 (2005), 4032–4039.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    N. Alon, A. Lubotzky and A. Wigderson: Semi Direct product in groups and zig-zag product in graphs: connections and applications, in: Proceedings of the 42nd Annual Symposium on the Foundations of Computer Science (FOCS), 630–637, 2001.

    Google Scholar 

  3. [3]

    S. Arora and M. Sudan: Improved low degree testing and its applications. Combinatorica 23 (2003), 365–426.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    L. Babai, L. Fortnow and C. Lund: Non-Deterministic Exponential Time has Two-Prover Interactive Protocols, Computational Complexity 1 (1991), 3–40.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    L. Babai, A. Shpilka and D. Stefankovic: Locally testable cyclic codes, IEEE Transactions on Information Theory 51 (2005), 2849–2858.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    E. Ben-Sasson, P. Harsha and S. Raskhodnikova: Some 3CNF Properties are Hard to Test, SIAM Journal on Computing 35 (2005), 1–21.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    E. Ben-Sasson and M. Sudan: Simple PCPs with poly-log rate and query complexity, STOC 2005 266–275, 2005.

    Google Scholar 

  8. [8]

    E. Ben-Sasson, M. Sudan, S. Vadhan and A. Wigderson: Randomnessefficient Low Degree Tests and Short PCPs via Epsilon-Biased Sets 35th Annual ACM Symposium, STOC 2003 612–621, 2003.

    Google Scholar 

  9. [9]

    S. D. Berman: Semisimple Cyclic and Abelian Codes, Cybernetics 3 (1967), 21–30.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    M. Blum, M. Luby and R. Rubinfeld: Self-Testing/Correcting with Applications to Numerical Problems, in: J. Comp. Sys. Sci. 47, 1993.

  11. [11]

    M. Capalbo, O. Reingold, S. Vadhan and A. Wigderson: Randomness Conductors and Constant-Degree Expansion Beyond the Degree /2 Barrier, Proceedings of the 34th STOC, 659–668, 2002.

    Google Scholar 

  12. [12]

    L. Carlitz and S. Uchiyama: Bounds for exponential sums, Duke Math. J. 24 (1957), 37–41.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    I. Dinur: The PCP theorem by gap amplification, J. ACM 54 (2007), 12.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    R. G. Gallager: Low density parity check codes, MIT Press, Cambridge, MA, 1963.

    Google Scholar 

  15. [15]

    E. Grigorescu, T. Kaufman and M. Sudan: Succinct Representation of Codes with Applications to Testing, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Springer Berlin Heidelberg, 2009. 534–547.

    Google Scholar 

  16. [16]

    O. Goldreich and M. Sudan: Locally testable codes and PCPs of almostlinear length, J. ACM 53 (2006), 558–655.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    T. Kaufman and M. Sudan: Algebraic Property Testing: The Role of Invariance, Proceedings of the 40th ACM Symposium on Theory of Computing (STOC) 2008.

    Google Scholar 

  18. [18]

    T. Kaufman and S. Litsyn: Almost Orthogonal Linear Codes are Locally Testable, FOCS 2005 (2005), 317–326.

    Google Scholar 

  19. [19]

    T. Kaufman and A. Lubotzky: Edge transitive ramanujan graphs and symmetric LDPC good codes, STOC 2012 359–366, 2012.

    Google Scholar 

  20. [20]

    M. Lackenby: Large groups, property (T) and the homology growth of subgroups, Math. Proc. Cambridge Philos. Soc. 146 (2009), 625–648.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    M. Lackenby: Covering spaces of 3-orbifolds, Duke Math. J. 136 (2007), 181–203.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    A. Lubotzky, B. Samuels and U. Vishne: Ramanujan complexes of type Ãd, Israel Journal of Mathematics 149 (2005), 267–299.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    A. Lubotzky, B. Samuels and U. Vishne: Explicit constructions of ramanujan complexes of type Ãd, Eur. J. Comb. 26 (2005), 965–993.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    A. Lubotzky and B. Weiss: Groups and expanders, in: Expanding Graphs (e. J. Friedman), DIMACS Ser. Discrete Math. Theoret. Compt. Sci. 95–109, Amer. Math. Soc., Prividence, RI 1993.

    Google Scholar 

  25. [25]

    M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi and D. A. Spielman: Improved Low-Density Parity-Check Codes Using Irregular Graphs, IEEE Transactions on Information Theory 47 (2001), 585–598.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    F. J. MacWilliams and N. J. A. Sloan: The Theory of Error Correcting Codes, North Holland, Amsterdam, 1977.

    Google Scholar 

  27. [27]

    R. J. McElice: On the Symmetry of Good Nonlinear Codes, IEEE Trans. Inform. Theory IT 16 (1970), 609–611.

    MathSciNet  Article  Google Scholar 

  28. [28]

    O. Meir: Combinatorial Construction of Locally Testable Codes, Proceedings of STOC 2008 285–294, 2008.

    Google Scholar 

  29. [29]

    R. Meshulam and A. Wigderson: Expanders in Group Algebras, Combinatorica 24 (2004), 659–680.

    MathSciNet  Article  MATH  Google Scholar 

  30. [30]

    J. Naor and M. Naor: Small-Bias Probability Spaces: Efficient Constructions and Applications, SIAM J. Comput. 22 (1993), 838–856.

    MathSciNet  Article  MATH  Google Scholar 

  31. [31]

    R. Rubinfeld and M. Sudan: Robust characterizations of polynomials with applications to program testing, SIAM Journal on Computing 25 (1996), 252–271.

    MathSciNet  Article  MATH  Google Scholar 

  32. [32]

    E. Rozenman, A. Shalev and A. Wigderson: A new family of Cayley expanders (?), 36th Annual ACM Symposium, STOC 2004 445–454, 2004.

    Google Scholar 

  33. [33]

    T. Richardson and R. Urbanke: The Capacity of Low-Density Parity Check Codes under Message-Passing Decoding, IEEE Transactions on Information Theory 47 (2001), 599–618.

    MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    O. Reingold, S. Vadhan and A. Wigderson: Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders, Annals of Mathematics 155 (2002), 157–187.

    MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    M. Sipser and D. A. Spielman: Expander codes, IEEE Transactions on Information Theory, 42 (1996), 1710–1722.

    MathSciNet  Article  MATH  Google Scholar 

  36. [36]

    M. Sudan: Lecture notes, http://people.csail.mit.edu/madhu/FT01/scribe/ bch.ps.

  37. [37]

    M. Sudan, L. Trevisan and S. Vadhan: Pseudorandom generators without the XOR Lemma, Journal of Computer and System Sciences 62 (2001), 236–266.

    MathSciNet  Article  MATH  Google Scholar 

  38. [38]

    R. M. Tanner: A recursive approach to low complexity codes, IEEE Transactions on Information Theory 27 (1981), 533–547.

    MathSciNet  Article  MATH  Google Scholar 

  39. [39]

    A. Weil: Sur les courbes algebriques et les varietes qui s'en deduisent Actualities Sci. et Ind. 1041, Hermann, Paris, 1948.

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Correspondence to Tali Kaufman.

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Kaufman, T., Wigderson, A. Symmetric LDPC codes and local testing. Combinatorica 36, 91–120 (2016). https://doi.org/10.1007/s00493-014-2715-1

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Mathematics Subject Classification (2000)

  • 05C25
  • 68P30
  • 68Q01