The above table shows E₄(n,d), the number of codewords in an optimal quaternary code of fixed length n and minimum edit distance d. For n ≤ 10, exact values are given where known, with upper and lower bounds provided when an exact value is not known. For n > 10, the value shown is the lower bound on E₄(n,d).

The earliest version of this table appeared in the technical report: S.K.Houghten, D.Ashlock and J.Lenarz, "Bounds on Optimal Edit-Metric Codes"and was ultimately published as "Construction of Optimal Edit Metric Codes", Proceedings of the 2006 IEEE Workshop on Information Theory (ITW 2006), 259-263.

Improvements since that time:

1. E₄(5,3) ≥ 44 and E₄(5,3) ≤ 46 (D.Ashlock & S.Houghten, July 2005)
2. E₄(6,4) ≤ 32 and thus E₄(7,4) ≤ 128, E₄(8,4) ≤ 512, E₄(9,4) ≤ 2048 and E₄(10,4) ≤ 8192 (D.Ashlock & S.Houghten, July 2005)
3. E₄(7,5) ≤ 28 and thus E₄(8,5) ≤ 112, E₄(9,5) ≤ 448 and E₄(10,5) ≤ 1792 (D.Ashlock & S.Houghten, July 2005)
4. E₄(5,3) = 44 and thus E₄(6,3) ≤ 176 (S.Houghten, August 2005)
5. E₄(7,5) ≤ 24 and thus E₄(8,5) ≤ 96, E₄(9,5) ≤ 384 and E₄(10,5) ≤ 1536 (S.Houghten, August 2006)
6. E₄(6,3) ≥ 108 (D.Ashlock and S.Houghten, October 2008)
7. Table expanded to n=16 (R.Flack & S.Houghten, 2008)
8. E₄(10,9) ≥ 5, E₄(11,7) ≥ 34 (and thus E₄(12,7) ≥ 34) , E₄(11,8) ≥ 14 (and thus E₄(12,8) ≥ 14), and E₄(11,9) ≥ 7 (and thus E₄(12,9) ≥ 7)(D.Ashlock & S.Houghten, March 2010)
9. E₄(6,3) ≥ 111, E₄(7,3) ≥ 343, E₄(8,3) ≥ 1082, E₄(7,4) ≥ 65, E₄(8,4) ≥ 173 (D.Ashlock, S.Houghten, J.A.Brown & J.Orth, September 2010)
10. E₄(9,4) ≥ 495, E₄(9,5) ≥ 97, E₄(9,6) ≥ 27, E₄(9,8) ≥ 5, E₄(10,6) ≥ 59, E₄(10,7) ≥ 20, E₄(11,5) ≥ 676, E₄(11,6) ≥ 133 (D.Ashlock, S.Houghten, J.A.Brown & J.Orth, September 2010)
11. E₄(11,3) ≥ 40604, E₄(11,4) ≥ 4574 (and thus E₄(12,4) ≥ 4574), E₄(12,5) ≥ 1894, E₄(13,5) ≥ 5517 (and thus E₄(14,5) ≥ 5517), E₄(11,6) ≥ 133, E₄(12,6) ≥ 338, E₄(13,6) ≥ 879 (and thus E₄(14,6) ≥ 879), E₄(12,7) ≥ 79, E₄(13,7) ≥ 188 (and thus E₄(14,7) ≥ 188), E₄(12,8) ≥ 28, E₄(13,8) ≥ 54 (and thus E₄(14,8) ≥ 54) and E₄(12,9) ≥ 12 (and thus E₄(13,9) ≥ 12) (D.Ashlock, S.Houghten, J.A.Brown & J.Orth, September 2010)
12. E₄(6,3) ≥ 114, E₄(7,3) ≥ 356, E₄(8,3) ≥ 1132, E₄(8,4) ≥ 176, E₄(10,5) ≥ 249, E₄(10,6) ≥ 60, E₄(11,6) ≥ 139, E₄(11,7) ≥ 41, E₄(11,8) ≥ 16, E₄(12,8) ≥ 30, E₄(12,9) ≥13 (and thus E₄(13,9) ≥ 13): see D.Ashlock, S.K.Houghten, J.A.Brown and J.Orth, On the synthesis of DNA error correcting codes, BioSystems 110 (2012), 1-8.
13. E₄(12,10) ≥ 6, E₄(13,9) ≥ 17, E₄(13,11) ≥ 5, E₄(14,9) ≥ 27, E₄(14,10) ≥ 13, E₄(14,11) ≥ 7, E₄(14,12) ≥ 5, E₄(15,9) ≥ 45, E₄(15,10) ≥ 19, E₄(15,11) ≥ 10, E₄(15,12) ≥6, E₄(16,9) ≥ 83, E₄(16,10) ≥ 32, E₄(16,11) ≥ 15, E₄(16,12) ≥ 8, and addition of values for E₄(17,d): see D.Ashlock and S.Houghten, Lexicode Crossover for Embeddable Biomarkers, IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology, 2015.
14. E₄(16,8) ≥ 196, E₄(16,11) ≥ 16, E₄(17,8) ≥ 322, E₄(17,9) ≥ 143, E₄(17,10) ≥ 57, E₄(17,11) ≥ 24, E₄(17,12) ≥ 13, and addition of values for E₄(n,13), D. Ashlock and S. Houghten, Hybridization and Ring Optimization for Larger Sets of Embeddable Biomarkers, 2017 IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology, 2017.

If you have improvements, comments, etc, please e-mail shoughten@brocku.ca