TY - GEN
T1 - Covering Codes for Insertions and Deletions
AU - Lenz, Andreas
AU - Rashtchian, Cyrus
AU - Siegel, Paul H.
AU - Yaakobi, Eitan
N1 - Funding Information: A. Lenz acknowledges support from the German-American Fulbright Commission for funding the visit to UCSD. E. Yaakobi acknowledges support For an integer q ≥ 2, let Σq⋃denote the q-ary alpha-from the Center for Memory and Recording Research at UCSD. This work bet {0, 1, . . . , q − 1} and Σq∗ = ℓ≥0 Σqℓ. We use len(x) researchand innovationprogramme(grantNo. 801434),byNSFgrant CCF-isalsofundedbytheEuropeanResearchCouncilundertheEU’sHorizon2020 to denote the length of x. For x = (x1, . . . , xn) ∈ Σqn, Funding Information: A. Lenz acknowledges support from the German-American Fulbright Commission for funding the visit to UCSD. E. Yaakobi acknowledges support from the Center for Memory and Recording Research at UCSD. This work is also funded by the European Research Council under the EU's Horizon 2020 research and innovation programme (grant No. 801434), by NSF grant CCFBSF- 1619053 and by the United States-Israel BSF grant 2018048. Publisher Copyright: © 2020 IEEE.
PY - 2020/6
Y1 - 2020/6
N2 - A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of insertions and deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single insertion or a single deletion that are tight up to a constant factor. Finally, we derive improved upper bounds for covering codes using R≥ 2 insertions or deletions. We prove that codes exist with density that is only a factor O(R log R) larger than the lower bounds for all fixed R. In particular, our upper bounds have an optimal dependence on the word length, and we achieve asymptotic density matching the best known bounds for Hamming distance covering codes.
AB - A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of insertions and deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single insertion or a single deletion that are tight up to a constant factor. Finally, we derive improved upper bounds for covering codes using R≥ 2 insertions or deletions. We prove that codes exist with density that is only a factor O(R log R) larger than the lower bounds for all fixed R. In particular, our upper bounds have an optimal dependence on the word length, and we achieve asymptotic density matching the best known bounds for Hamming distance covering codes.
UR - https://www.scopus.com/pages/publications/85090413655
U2 - 10.1109/ISIT44484.2020.9174295
DO - 10.1109/ISIT44484.2020.9174295
M3 - Conference contribution
T3 - IEEE International Symposium on Information Theory - Proceedings
SP - 723
EP - 728
BT - 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings
T2 - 2020 IEEE International Symposium on Information Theory, ISIT 2020
Y2 - 21 July 2020 through 26 July 2020
ER -