\(q\)-ary linear LTC 

Description

A \(q\)-ary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\). More technically, the code is a \((u,R)\)-LTC if the rows of its parity-check matrix \(H\in GF(q)^{r\times n}\) have weight at most \(u\) and if \begin{align} \frac{1}{r}|H x| \geq \frac{R}{n} D(x,C) \tag*{(1)}\end{align} holds for any \(q\)-ary string \(x\), where \(D(x,C)\) is the \(q\)-ary Hamming distance between \(x\) and the closest codeword to \(x\) [1; Def. 11].

Parents

Children

  • Ben-Sasson-Sudan code
  • Meir code — Meir codes stand out in that they are based on a combinatorial construction, while other LTCs often use algebraic tools.

Cousins

  • Generalized RM (GRM) code — GRM codes for \(r<q\) can be LTCs in the low- [2,3] and high-error [4,5] regimes.
  • Classical Goppa code — Goppa codes are locally testable [6].
  • Reed-Solomon (RS) code — RS codes can be used to construct LTCs encoding \(k\) bits with length \(k \text{polylog}(k)\) and query complexity \(\text{polylog}(k)\) [7].
  • Bose–Chaudhuri–Hocquenghem (BCH) code — Duals of BCH codes are locally testable [6].
  • Cyclic linear \(q\)-ary code — Cyclic linear codes cannot be \(c^3\)-LTCs [8]. Codeword symmetries are in general an obstruction to achieving such LTCs [9].
  • Linear \(q\)-ary code — Linear \(q\)-ary codes with distances \(\frac{1}{2}n-\sqrt{t n}\) for some \(t\) are called almost-orthogonal and are locally testable with query complexity of order \(O(t)\) [6]. This was later improved to codes with distance \(\frac{1}{2}n-O(n^{1-\gamma})\) for any positive \(\gamma\) [10], provided that the number of codewords is polynomial in \(n\).
  • Expander LP code — Classical codes resulting from the expander lifted-product construction are one of the first two families of \(c^3\)-LTCs.

References

[1]
A. Leverrier, V. Londe, and G. Zémor, “Towards local testability for quantum coding”, Quantum 6, 661 (2022) arXiv:1911.03069 DOI
[2]
L. Babai et al., “Checking computations in polylogarithmic time”, Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC ’91 (1991) DOI
[3]
S. Arora et al., “Proof verification and the hardness of approximation problems”, Journal of the ACM 45, 501 (1998) DOI
[4]
R. Raz and S. Safra, “A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP”, Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC ’97 (1997) DOI
[5]
S. Arora and M. Sudan, “Improved low-degree testing and its applications”, Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC ’97 (1997) DOI
[6]
T. Kaufman and S. Litsyn, “Almost Orthogonal Linear Codes are Locally Testable”, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) DOI
[7]
E. Ben-Sasson and M. Sudan, “Short PCPs with Polylog Query Complexity”, SIAM Journal on Computing 38, 551 (2008) DOI
[8]
L. Babai, A. Shpilka, and D. Stefankovic, “Locally Testable Cyclic Codes”, IEEE Transactions on Information Theory 51, 2849 (2005) DOI
[9]
M. Sudan, “Invariance in Property Testing”, Property Testing 211 (2010) DOI
[10]
T. Kaufman and M. Sudan, “Sparse Random Linear Codes are Locally Decodable and Testable”, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07) (2007) DOI
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Zoo Code ID: q-ary_ltc

Cite as:
\(q\)-ary linear LTC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_ltc
BibTeX:
@incollection{eczoo_q-ary_ltc,
  title={\(q\)-ary linear LTC},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/q-ary_ltc}
}
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Cite as:

\(q\)-ary linear LTC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_ltc

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/ltc/q-ary_ltc.yml.