We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. Moreover, we show that the produced classical LDPC codes are also asymptotically good and locally testable with constant query and soundness parameters, which proves a well-known conjecture in the field of locally testable codes.

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We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.

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We consider a convolution operator in R d with kernel in L q acting from L p to L s , where 1/p + 1/q = 1 + 1/s. The main theorem states that if 1 < q, p, s < ∞, then there exists an L p function of unit norm on which the s-norm of the convolution is attained. A number of questions, related to the statement and proof of the main theorem, are discussed. Also the problem of computing best constants in the Hausdorff-Young inequality for the Laplace transform, which prompted this research, is considered.1 Emphasizing that K p,r is the extremum of the symmetric bilinear form K p,r = K r,p = sup k(x + y)f (x)g(y) dx dy ; f p = g r = 1. 2 The labels (a) , (b) etc. refer to the comments (a), (b) etc. in Section 8. 1. The class Max comprises all maximizing sequences (for the convolution operator K : L p → L r ). 2. The class SMax comprises all special maximizing sequences, that is, maximizing sequences of the form (Bf n ), where (f n ) ∈ Max. 3. The class RTgt comprises all relatively tight sequences. 4. The class Tgt comprises all tight sequences. 5. The class WCvg comprises all weakly convergent sequences. 6. The class LCvg comprises all locally convergent sequences, i.e. sequences converging in L p norm on any bounded measurable subset of R d .