タイトル：Constructions of ℓ-Adic t-Deletion-Correcting Quantum Codes
著者名：Ryutaroh MATSUMOTO; Manabu HAGIWARA
雑誌名：IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 号数：Volume E105.A, Issue 3 DOI：https://doi.org/10.1587/transfun.2021EAP1034
Abstract：We propose two systematic constructions of deletion-
correcting codes for protecting quantum inforomation. The first one
works with qudits of any dimension ℓ, which is referred to as ℓ-adic,
but only one deletion is corrected and the constructed codes are
asymptotically bad. The second one corrects multiple deletions and can
construct asymptotically good codes. The second one also allows
conversion of stabilizer-based quantum codes to deletion-correcting
codes, and entanglement assistance.
タイトル： Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions 著者名： Ryan Gabrys, Venkatesan Guruswami, Joao Ribeiro, Ke Wu アクセス： https://arxiv.org/pdf/2112.09971.pdf
Abstract： We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, who proved that binary Varshamov-Tenengolts codes correct one arbitrary edit error, i.e., one deletion or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations…
Abstract： Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound. Specially, a class of optimal codebooks with respect to the Levenshtein bound is obtained. These classes of codebooks are constructed by selecting certain rows deterministically from circulant matrices, Fourier matrices and Hadamard matrices, respectively. The construction methods and parameters of some codebooks provided in this paper are new.