In this paper, we present an efficient strategy to enumerate the number of k-cycles, g≤k<2g, in the Tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth g using its polynomial parity-check matrix H. This strategy works for both (dv,dc)-regular and irregular QC-LDPC codes. In this approach, we note that the mth power of the polynomial adjacency matrix can be used to describe walks of length m in the protograph and can therefore be sufficiently described by the matrices Bm(H)(HHT)m/2H(m2), where m≥0. We provide formulas for the number of k-cycles, Nk, by just taking into account repetitions in some multisets constructed from the matrices Bm(H). This approach is shown to have low complexity. For example, in the case of QC-LDPC codes based on the 3×nv fully-connected protograph, the complexity of determining Nk, for k=4,6,8,10 and 12, is O(nv2log(N)), O(nv2log(nv)log(N)), O(nv4log4(nv)log(N)), O(nv4log(nv)log(N)) and O(nv6log6(nv)log(N)), respectively. The complexity, depending logarithmically on the lifting factor N, gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes.
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Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields
For every prime p > 0, every n > 0 and κ = O(log n), we show the existence of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over 𝔽_p with M extension variables, each depending on at most κ original variables requires size exp(Ω(n²)/10^κ(M + n log n))
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- Award ID(s):
- 2212135
- PAR ID:
- 10531032
- Editor(s):
- Ta-Shma, Amnon
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 264
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-282-2
- Page Range / eLocation ID:
- 264-264
- Subject(s) / Keyword(s):
- Proof complexity Algebraic proof systems Polynomial Calculus Extension variables AC⁰[p]-Frege Theory of computation → Proof complexity
- Format(s):
- Medium: X Size: 24 pages; 854548 bytes Other: application/pdf
- Size(s):
- 24 pages 854548 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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In this paper, we present an efficient strategy to enumerate the number of k-cycles, g ≤ k < +2g, in the Tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth g using its polynomial parity-check matrix H. This strategy works for both (n c , n v )-regular and irregular QC-LDPC codes. In this approach, we note that the mth power of the polynomial adjacency matrix can be used to describe walks of length m in the protograph and can therefore be sufficiently described by the matrices Bm(H)≜(HH⊤)⌊m/2⌋H(mmod2), where m ≥ 0. For example, in the case of QC-LDPC codes based on the 3 × n v fully-connected protograph, the complexity of determining the number of k-cycles, Nk, for k = 4, 6 and 8, is O(n2vlog(N)), O(n2vlog(nv)log(N)) and O(n4vlog4(nv)log(N)), respectively. The complexity, depending logarithmically on the lifting factor N, gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.CCC.2023.7
