Pith. sign in

REVIEW 3 major objections 4 minor 55 references

Constructions of Covering Sequences and Arrays

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that covering sequences can be built within a constant factor of the sphere-covering bound, and it improves the known upper bounds for all short lengths up to 20.

desk verdict Worth publishing after fixing an overreaching abstract, a parity typo in Theorem 24, and providing verification data for the computer-assisted table entries. read the letter →

arxiv 2502.08424 v2 pith:5BS3EIA5 submitted 2025-02-12 math.CO

Yeow Meng Chee , Tuvi Etzion , Hoang Ta , Van Khu Vu This is my paper · ORCID
classification math.CO MSC 94B6594B25
keywords coveringsequencescodessphere-coveringboundcyclicsequenceself-dualHammingcode2D-sequencesdeBruijngraph
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

A covering sequence is a cyclic binary string whose consecutive length-n windows form a covering code of radius R: every n-bit word is within Hamming distance R of some window. The paper tries to show that such sequences can be nearly as short as the sphere-covering lower bound, and to improve the known shortest lengths for every n from 9 to 20 with radius 1, 2, or 3. Its main asymptotic result is that for radius 1, merging the codewords of the cyclic Hamming code gives a covering sequence within a factor of 2 of optimal, and a construction based on self-dual sequences brings the factor down to 1.25. The same tools are extended to covering sequence codes and to two-dimensional covering arrays whose m-by-n windows cover all matrices.

What carries the argument

The load-bearing objects are cyclic covering sequence codes (CSCs): sets of cyclic words whose length-n windows jointly form an $(n,R)$-covering code. Given a CSC, the paper converts each cyclic codeword into an acyclic word by appending its first $n-1$ bits, then merges all words into one cycle by maximizing prefix-suffix overlaps, using a set-cover approximation for the shortest cyclic superstring problem. The asymptotic bounds come from two specific CSCs: the cyclic Hamming code for $n=2^k-1$, and a recursively defined family of self-dual sequences of the form $[X\bar{X}]$ for $n=2^k$, whose covering property is delegated to the companion theory of nearly-perfect covering codes. For 2D arrays, the machinery is folding a one-dimensional covering sequence into rows and arranging many cyclic shifts of a covering sequence, guided by de Bruijn sequences of shift indices.

What would settle it

Recompute the claimed (16,1)-covering sequence of length 4462: regenerate the 64 self-dual words of Appendix D from Construction 1, merge them with the stated overlap total of 594, and test every 16-bit word against the merged cycle to see whether any word lies at distance greater than 1 from all windows; the same test can be run on the 3516-bit (15,1)-covering sequence of Appendix C.

Watch

Extended reading notes

Core claim

In the authors' terms, the paper's central claim is that covering sequences can be constructed whose length is only a constant factor above the sphere-covering bound, and that several previously known upper bounds for small parameters can be improved by explicit constructions. For radius 1, Section III reduces the binary Hamming code of length $2^k-1$ to its distinct cyclic shifts and merges them by maximizing overlaps, producing a $(2^k-1,1)$-covering sequence shorter than $2^{2^k-k}$, within a factor of at most 2 of the sphere-covering bound. Section IV uses self-dual sequences of the form $[X\bar{X}]$ to build a $(2^k,1)$-covering sequence code whose total length equals the size of an optimal $(2^k,1)$-covering code, and then merges its codewords to obtain a $(2^k,1)$-CS of length less than $1.25K$, where $K=2^{2^k-k}$. The paper also generalizes covering sequences to $(n,m,R)$-covering sequence codes and to $(m\times n,R)$-covering 2D-sequences, giving both a folding construction and a shift-based construction for the latter, and it tabulates improved upper bounds on $L(n,R)$ for $9\le n\le 20$ and $1\le R\le 3$.

Load-bearing premise

The improved small-case upper bounds rest on computer-generated codeword lists and overlap counts printed in the appendices, with no verifier shipped, so a single wrong overlap or non-covering codeword would change a bound; the factor-1.25 asymptotic result likewise depends on the self-dual covering-code family whose proof is delegated to reference [6].

Editorial extensions

If this is right

  • For binary length $n=2^k-1$, covering sequences exist with length below $2^{2^k-k}$, so the gap to the sphere-covering bound is at most a factor of 2.
  • For $n=2^k$, covering sequences exist within a factor of 1.25 of the optimal covering-code size, giving the closest known asymptotic match for radius 1.
  • The table of bounds for $9\le n\le 20$ and $1\le R\le 3$ improves several upper bounds on the shortest covering sequence length, with entries coming from interleaving, self-dual merges, and primitive-polynomial constructions.
  • Folding a good one-dimensional covering sequence yields $m\times n$ covering arrays of area roughly twice the sequence length, while arranging shifts of a covering sequence can produce arrays whose area is smaller than the corresponding one-dimensional covering sequence.
  • The interleaving construction composes an $(n_1,R_1)$-covering sequence and an $(n_2,R_2)$-covering sequence into an $(n_1+n_2,R_1+R_2)$-covering sequence of length $2k_1k_2$, which is what produces most of the larger-radius entries in the table.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the delegated self-dual proof in reference [6] holds, the same recursive construction may be adaptable to radii 2 and 3 using nearly-perfect or Preparata-type codes, a direction the authors list as future work.
  • The shortest-cyclic-superstring merge step is heuristic; replacing the set-cover approximation with an exact solver or a tighter bound on overlaps could lower several table entries without changing the underlying covering codes.
  • The observation that shift-arranged 2D arrays can be smaller in area than the best known one-dimensional covering sequence suggests that 2D covering arrays deserve their own lower bounds rather than being compared only through folding.
  • For the self-dual construction, the factor 1.25 is obtained by concatenating without computing overlaps; actually computing overlaps—as is done for the small cases in the appendices—would likely reduce the constant further.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies (n,R)-covering sequences, their multi-sequence generalization (n,m,R)-covering sequence codes, and a two-dimensional generalization (m×n,R)-covering 2D-sequences. It presents several construction methods: merging codewords of a cyclic covering code through shortest cyclic superstrings (Section III), a self-dual-sequence construction (Section IV), interleaving constructions (Section V), a primitive-polynomial construction (Section VI), and folding/shift constructions for 2D-sequences (Section VII). The paper reports improved upper bounds in Table I for 9≤n≤20 and 1≤R≤3, with explicit codeword lists and overlap computations in the appendices. The asymptotic statements in the body give a Hamming-code based (2^k−1,1)-CS within a factor 2 of the sphere-covering bound and a self-dual based (2^k,1)-CS within a factor 1.25, whereas the abstract claims a general constant-factor asymptotic result for covering sequences.

Significance. The deterministic Hamming-code and self-dual constructions are genuine strengths: for radius one they provide two infinite families whose lengths are within constant factors of the sphere-covering bound, improving on the general O(q^n/V(n,R) log n) upper bound for those parameters. The small-parameter upper bounds and the 2D generalizations are useful contributions, and the codeword lists in the appendices are checkable in principle. However, the main asymptotic claim as stated in the abstract is not supported by the body beyond radius one, and the computational upper bounds are not accompanied by a verifier or code, so the significance of the headline claims is lower than the abstract suggests.

major comments (3)
  1. [Abstract and Sections III–VIII] The abstract states: 'We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.' As written, this has no restriction on R. The body proves this only for R=1 and only for the families n=2^k−1 and n=2^k (Section III gives a factor-2 bound and Section IV a factor-1.25 bound for these two families). The interleaving constructions in Section V have lengths that are products or quadratic forms of the input lengths, the primitive-polynomial construction in Section VI gives an exponential ratio for fixed R≥2, and the 2D results in Section VII retain a logarithmic factor (Proposition 2). Thus the unrestricted asymptotic claim is unsupported. The abstract and the conclusion should be qualified to radius 1, or to the specific n families, and the general-R case should be stated as an open problem.
  2. [Section III and Appendices A–D] The improved upper bounds in Table I marked 'computer search' rely on the shortest-cyclic-superstring merging described in Section III and on the codeword lists and overlap totals asserted in Appendices A–D. No code, verifier script, or machine-readable certificate is provided, so a single erroneous overlap or non-covering codeword would change an upper bound. Given that these upper bounds are a central contribution, the authors should make the verification reproducible, for example by supplying the search code or a verifier that checks each listed sequence for the claimed covering property and each overlap sum.
  3. [Section VII, Theorem 24] Theorem 24 states the parity condition in terms of n ('If n is even... If n is odd...'), but Construction 5 and the proof of Theorem 24 both make the number of rows depend on k, the length of the input sequence, not on the window length n. As printed, the theorem statement is incorrect; the parity condition should refer to k. This is a local error, but it must be corrected because the theorem as stated would mislead a reader applying Construction 5.
minor comments (4)
  1. [Section VI, Lemma 13] The proof of Lemma 13 is not written out convincingly: the sentence 'This follows from the fact that after n consecutive ones we should have a zero in such a sequence' is not an argument. The claim follows immediately from c(1) != 0 for a primitive polynomial, so the proof should be replaced by that short argument.
  2. [Section VI, Theorem 17] The construction in Theorem 17 is stated for 'n + 2R + 1' but the surrounding text sometimes refers to 'span n' and 'length 2^n−1' without consistently distinguishing the window length from the recurrence order. Please clarify notation in this section.
  3. [Section V, Example 12] The example says a (10,1)-CS of length 177 yields a (20,2)-CS of length 178·178, but the preceding paragraph says Construction 3 gives k(k+1) or (k+1)^2 depending on parity; the example uses 178^2 for k=177? Please check the parity cases and make the formula and example agree.
  4. [Appendices A-D] Several appendix entries have minor typographical issues, such as 'The following sequences is a ...' and inconsistent spacing in the codeword lists. These should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constructions are explicit and the bounds are derived by direct comparison to the sphere-covering bound.

full rationale

The derivation chain is not circular. Section III constructs (2^k-1,1)-CSs from the cyclic Hamming code: the bound "length shorter than 2^{2^k-k}" is an explicit upper bound obtained by merging the 2^{n-k} codewords after adding at most n-1 bits each, compared against the sphere-covering lower bound 2^{2^k-1-k}; the factor 2 is a direct ratio, not a fit. Section IV's factor-1.25 bound similarly compares an explicit concatenation length 2^{2^k-2k-2}(2^{k+2}+2^k-1) with the sphere bound K=2^{2^k}/(2^k+1); Theorem 7's correctness is delegated to the overlapping-author paper [6], but [6] is an external published theorem about nearly-perfect covering codes, not a restatement of the present claim, and no parameter is fitted to the target quantity. Sections V and VI prove interleaving and primitive-polynomial constructions with self-contained arguments. The abstract's phrase "up to a constant factor" is broader than the radius-1 families actually proved; this is an overclaim/correctness concern, not circularity. The computer-search upper bounds in Table I and Appendices A-D lack a shipped verifier, but that is a reproducibility issue, not a circular reduction. Proposition 2 is imported from the authors' conference version [9] and from Vu [54], but these are prior existence theorems with independent statements and the paper explicitly says it will not reprove them; reliance on prior theorems is not circularity. No self-definitional or fitted-input-as-prediction step occurs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central constructions rest on standard Hamming/de Bruijn facts and on two external existence results [6], [9] with overlapping authors; there are no fitted parameters. The computer-search table adds an empirical premise without shipped verification code.

assumptions (5)
  • standard math The Hamming code of length 2^k-1 is perfect, cyclic, and every codeword orbit has size dividing n with few degenerate orbits.
    Used in Section III to count cyclic Hamming codewords and derive a (2k-1,1)-CS within factor 2 of optimal.
  • domain assumption The self-dual (n,2n,1)-CSC family of Construction 1 exists and has the stated size and uniqueness property, proved in [6].
    Section IV's 1.25-factor (2k,1)-CS rests on this external construction by overlapping authors.
  • domain assumption There exists a primitive polynomial c(x) with c_i=0 for 1 <= i <= 2R+1 for the relevant n and R.
    Lemma 16 and Theorem 17 need such polynomials; the paper asserts existence for small and larger n without a proof.
  • domain assumption The computer-searched CSCs and the overlap totals in Appendices A-D are correct.
    Table I upper bounds from computer search depend on these unverified lists; no code or verifier is shipped.
  • standard math L(n,R) <= L(n+1,R) (Theorem 3), and span-n de Bruijn sequences exist.
    Used in Table I row 'h' and in Construction 6 for de Bruijn sequences over Z_k.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Constructions of Covering Sequences and Arrays." pith.science (2026). https://pith.science/paper/5BS3EIA5

@misc{pith2026250208424,
  author       = {Pith},
  title        = {Pith review of: Constructions of Covering Sequences and Arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5BS3EIA5}},
  note         = {Machine review of arXiv:2502.08424}
}
abstract

An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. Using several construction methods improvements of the upper bounds on the length of such sequences for $n \leq 20$ and $1 \leq R \leq 3$, are obtained. The definition is generalized in two directions. An $(n,m,R)$-covering sequence code is a set of cyclic sequences of length $m$ whose consecutive $n$-tuples form a code of length~$n$ and covering radius $R$. The definition is also generalized to arrays in which the $m \times n$ sub-matrices form a covering code with covering radius $R$. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [6]

    B ORUCHOVSKY , T

    A. B ORUCHOVSKY , T. E TZION , AND R. M. R OTH, On nearly perfect covering codes, IEEE Trans. on Infor. Theory, 71 (2025) 2494– 2504

    work page 2025

  2. [1]

    B ERKOWITZ AND S

    R. B ERKOWITZ AND S. K OPPARTY, Robust positioning patterns, in Proc. 27th Annu. ACM-SIAM Disc. Algorithms (2016) 1937–1951

    work page 2016

  3. [2]

    B LAUM AND J

    M. B LAUM AND J. B RUCK , MDS array codes for correcting criss-cross errors, IEEE Trans. on Infor. Theory, 46 (2000) 1068–1077

    work page 2000

  4. [3]

    B LAUM , J

    M. B LAUM , J. B RUCK , A. VARDY, Interleaving schemes for multidimensional cluster errors, IEEE Trans. on Infor. Theory, 44 (1998) 730–743

    work page 1998

  5. [4]

    B LUM , T

    A. B LUM , T. J IANG , M. L I, J. T ROMP, AND M. YANNAKAKIS , Linear approximation of shortest superstrings, Journal Of The ACM (JACM). 41 (1994) 630–647

    work page 1994

  6. [5]

    B REITBACH , M

    M. B REITBACH , M. B OSSERT , V. ZYABLOV, V. SIDORENKO , Array codes correcting a two-dimensional cluster of errors, IEEE Trans. on Infor. Theory, 44 (1998) 2025–2031

    work page 1998

  7. [7]

    A. M. B RUCKSTEIN , T. E TZION , R. G IRYES , N. G ORDON , R. J. H OLT, AND D. S HULDINER , Simple and robust binary self-location patterns, IEEE Trans. on Infor. Theory, 58 (2012) 4884–4889

    work page 2012

  8. [8]

    Y. M. C HEE , D. T. D AO, H. M. K IAH , S. L ING , AND H. W EI, Robust positioning with low redundancy, SIAM J. Comput. 49 (2020) 284–317

    work page 2020

Show all 55 references
  1. [9]

    Y. M. C HEE , T. E TZION , H. T A, AND V. K. V U, On de Bruijn Covering Sequences and Arrays, in Proceedings IEEE Symposium on Information Theory, Athens, Greece 2024, pp. 1343–1348

    2024

  2. [10]

    C HUNG AND J

    F. C HUNG AND J. N. C OOPER , De bruijn cycles for covering codes, Random Structures & Algorithms, 25 (2004) 421—431

    2004

  3. [11]

    C OHEN , I

    G. C OHEN , I. H ONKALA , S. L ITSYN , AND A. L OBSTEIN , Covering Codes, North-Holland, Amsterdam, 1997

    1997

  4. [12]

    G. D. C OHEN , M. G. K ARPOVSKY , H. F. M ATTSON JR., AND J. R. S CHATZ , Covering radius - survey and recent results, IEEE Trans. on Infor. Theory, 31 (1985) 328–343

    1985

  5. [13]

    G. D. C OHEN , A. C.L OBSTEIN , AND N. J. A. S LOANE , Further results on the covering radius of codes, IEEE Trans. on Infor. Theory, 32 (1986) 680–694

    1986

  6. [14]

    C. J. C OLBOURN , Combinatorial aspects of covering arrays, Le Matematiche, 59 (2004) 125—172

    2004

  7. [15]

    C. J. C OLBOURN , G. K ERI , P. P. S ORIANO , AND J.-C. S CHLAGE -P UTCH , Covering and radius-covering arrays: constructions and classification, Disc. Math, 158 (2010) 1158–1180

    2010

  8. [16]

    D OUGHERTY AND H

    R. D OUGHERTY AND H. JANWA, Covering radius computations for binary cyclic codes, Mathematics of computation, 57 (1991) 415– 434

    1991

  9. [17]

    D OWNIE AND N

    D. D OWNIE AND N. S LOANE , The covering radius of cyclic codes of length up to 31,IEEE Trans. on Infor. Theory, 31 (1985) 446–447

    1985

  10. [18]

    E TZION , Constructions for perfect maps and pseudorandom arrays, IEEE Trans

    T. E TZION , Constructions for perfect maps and pseudorandom arrays, IEEE Trans. on Infor. Theory, 34 (1988) 1308–1316

    1988

  11. [19]

    E TZION , Sequences and the de Bruijn Graph: Properties, Constructions, and Applications, London, UK; San Diego, US, Cambridge, US: Elsevier, 2024

    T. E TZION , Sequences and the de Bruijn Graph: Properties, Constructions, and Applications, London, UK; San Diego, US, Cambridge, US: Elsevier, 2024

    2024

  12. [20]

    E TZION , Pseudo-Random and de Bruijn Array Codes, IEEE Int

    T. E TZION , Pseudo-Random and de Bruijn Array Codes, IEEE Int. Symp. on Inf. Theory , pp. 1742–1747, Athens, Greece, July 2024

    2024

  13. [21]

    E TZION , On de Bruijn array codes, Part I: nonlinear codes, IEEE Trans

    T. E TZION , On de Bruijn array codes, Part I: nonlinear codes, IEEE Trans. on Infor. Theory, 71 (2025) 1434–1449

    2025

  14. [22]

    E TZION AND G

    T. E TZION AND G. G REENBERG , Constructions for perfect mixed Codes and other covering codes, IEEE Trans. on Infor. Theory, 39 (1993) 209–214

    1993

  15. [23]

    E TZION AND A

    T. E TZION AND A. L EMPEL , Construction of de bruijn sequences of minimal complexity, IEEE Trans. on Infor. Theory, 30 (1984) 705–709

    1984

  16. [24]

    E TZION AND B

    T. E TZION AND B. M OUNITS Quasi-perfect codes with small distance, IEEE Trans. on Infor. Theory 51 (2005) 3938–3946

    2005

  17. [25]

    E TZION AND K

    T. E TZION AND K. G. P ATERSON , Near optimal single-track gray codes, IEEE Trans. on Infor. Theory, 42 (1996) 779–789. 22

    1996

  18. [26]

    E TZION AND A

    T. E TZION AND A. VARDY, Two-dimensional interleaving schemes with repetitions: Constructions and bounds, IEEE Trans. on Infor. Theory, 48 (2002) 428–457

    2002

  19. [27]

    E TZION AND E

    T. E TZION AND E. YAAKOBI , Error-correction of multidimensional bursts, IEEE Trans. on Infor. Theory, 55 (2009) 961–976

    2009

  20. [28]

    G ALLANT , D

    J. G ALLANT , D. M AIER , D., AND J. A STORER , On finding minimal length superstrings, Journal Of Computer And System Sciences , 20 (1980) 50–58

    1980

  21. [29]

    G INGERAS , J

    T. G INGERAS , J. M ILAZZO , D. S CIAKY , AND R. R OBERTS , Computer programs for the assembly of DNA sequences, Nucleic Acids Research, 7 (1979) 529–543

    1979

  22. [30]

    S. W. G OLOMB , Shift Register Sequences, Singapore: World Scientific, 2017

    2017

  23. [31]

    R. L. G RAHAM AND N. J. A. S LOANE , On the covering radius of codes, IEEE Trans. on Infor. Theory, 31 (1985) 385–401

    1985

  24. [32]

    H ORAN AND B

    V. H ORAN AND B. S TEVENS , Locating patterns in the de Bruijn torus, Disc. Math. 339 (2016) 1274–1282

    2016

  25. [33]

    J ANWA, Some new upper bounds on the covering radius of binary linear codes, IEEE Trans

    H. J ANWA, Some new upper bounds on the covering radius of binary linear codes, IEEE Trans. on Infor. Theory, 35 (1989) 110–112

    1989

  26. [34]

    J IANG AND M

    T. J IANG AND M. L I, Approximating shortest superstrings with constraints, Theoretical Computer Science . 134 (1994) 473–491

    1994

  27. [35]

    K APLAN AND N

    H. K APLAN AND N. S HAFRIR , The greedy algorithm for shortest superstrings, Information Processing Letters . 93 (2005) 13–17

    2005

  28. [36]

    K AVUT AND S

    S. K AVUT AND S. T UTDERE , The covering radii of a class of binary cyclic codes and some bch codes, Des., Codes and Crypt., 87 (2019) 317–325

    2019

  29. [37]

    K RIVELEVICH , B

    M. K RIVELEVICH , B. S UDAKOV, AND V. H. V U, Covering codes with improved density, IEEE Trans. on Infor. Theory, 49 (2003) 1812–1815

    2003

  30. [38]

    P. V. K UMAR AND V. K. W EI, Minimum distance of logarithmic and fractional partial m-sequences, IEEE Trans. on Infor. Theory, 38 (1992) 1474–1482

    1992

  31. [39]

    F. J. M ACWILLIAMS AND N. J. A. S LOANE , Pseudo-random sequences and arrays, Proceedings of the IEEE, 64 (1976) 1715–1729

    1976

  32. [40]

    C. J. M ITCHELL , Aperiodic and semi-periodic perfect maps, IEEE Trans. on Infor. Theory, 41 (1995) 88–95

    1995

  33. [41]

    C. J. M ITCHELL AND P. R. W ILD , Constructing orientable sequences, IEEE Trans. on Infor. Theory, 68 (2022) 4782–4789

    2022

  34. [42]

    R. A. M ORANO , C. O ZTURK , R. C ONN , S. D UBIN , S. Z IETZ , AND J. N ISSANO , Structured light using pseudorandom codes, IEEE Transactions on Pattern Analysis and Machine Intelligence, 20 (1998) 322–327

    1998

  35. [43]

    K. G. P ATERSON , Perfect maps, IEEE Trans. on Infor. Theory, 40 (1994) 743–753

    1994

  36. [44]

    C. D. R OSIN , Using reasoning models to generate search heuristics that solve open instances of combinatorial design problemss, https://arxiv.org/abs/2505.23881 (2025)

    2025 arXiv

  37. [45]

    R. M. R OTH, Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. on Infor. Theory, 37 (1991) 328–336

    1991

  38. [46]

    S CHWARTZ AND J

    M. S CHWARTZ AND J. B RUCK , Constrained codes as networks of relations, IEEE Trans. on Infor. Theory, 54 (2008) 2179–2195

    2008

  39. [47]

    N. J. A. S LOANE , Covering arrays and intersection codes, J. of Combin. Designs, 1 (1993) 51–63

    1993

  40. [48]

    S TEPHEN , String searching algorithms, World Scientific, 1994

    G. S TEPHEN , String searching algorithms, World Scientific, 1994

    1994

  41. [49]

    S TORER , Data compression: methods and theory, Computer Science Press, Inc., 1987

    J. S TORER , Data compression: methods and theory, Computer Science Press, Inc., 1987

    1987

  42. [50]

    S TRUIK , Covering codes, Ph.D

    R. S TRUIK , Covering codes, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1994

    1994

  43. [51]

    I. T AL, T. E TZION , AND R. R OTH, On row-by-row coding for 2d constraints, IEEE Trans. on Infor. Theory, 55 (2009) 3565–3576

    2009

  44. [52]

    W ANG , S

    L. W ANG , S. H U, AND O. S HAYEVITZ , Quickest sequence phase detection, IEEE Trans. Infor. Theory 63 (2017) 5834–5849

    2017

  45. [53]

    V AZIRANI , Approximation algorithms

    V. V AZIRANI , Approximation algorithms. Springer, 2001

    2001

  46. [54]

    V U, De bruijn covering codes with arbitrary alphabets, Advances in Applied Mathematics, 34 (2005) 65–70

    V. V U, De bruijn covering codes with arbitrary alphabets, Advances in Applied Mathematics, 34 (2005) 65–70

    2005

  47. [55]

    W EI, Nearly optimal robust positioning patterns, IEEE Trans

    H. W EI, Nearly optimal robust positioning patterns, IEEE Trans. Infor. Theory 68 (2022) 193–203

    2022

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.