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Two series of equitable symbol weight codes meeting the Plotkin bound

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Abstract

A \(q\)-ary code of length \(n\) is termed an equitable symbol weight code, if each symbol appears among the coordinates of every codeword either \(\lfloor n/q \rfloor \) or \(\lceil n/q \rceil \) times. This class of codes was proposed recently by Chee et al. in order to more precisely capture a code’s performance against permanent narrowband noise in power line communication. In this paper, two series of new equitable symbol weight codes of optimal sizes meeting the Plotkin bound are constructed via combinatorial designs.

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References

  1. Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).

  2. Blake I.F., Mullin R.C.: An Introduction to Algebraic and Combinatorial Coding Theory. Academic Press, New York (1976).

  3. Bogdanova G., Todorov T., Zinoviev V.A.: On construction of \(q\)-ary equidistant codes. Probl. Inf. Transm. 43, 13–36 (2007).

    Google Scholar 

  4. Chee Y.M., Kiah H., Ling A., Wang C.: Optimal equitable symbol weight codes for power line communications. In: ISIT 2012—Proceedings of the 2012 IEEE International Symposium on Information Theory, Cambridge, pp. 671–675 (2012).

  5. Chee Y.M., Kiah H., Purkayastha P., Wang C.: Importance of symbol equity in coded modulation for power line communications. In: ISIT 2012—Proceedings of the 2012 IEEE International Symposium on Information Theory, Cambridge, pp. 666–670 (2012).

  6. Chee Y.M., Kiah H., Wang C.: Generalized balanced tournament designs with block size four, Preprint.

  7. Colbourn C.J., Dinitz J.H.: The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton (2007).

  8. Colbourn C.J., Lamken E.R., Ling A.C.H., Mills W.H.: The existence of Kirkman squares-doubly resolvable \((v, 3, 1)\)-BIBDs. Des. Codes Cryptogr. 26, 169–196 (2002).

  9. Furino S.C., Miao Y., Yin J.: Frames and Resolvable Designs. CRC Press, Boca Raton (1996).

  10. Lamken E.R.: Generalized balanced tournament designs. Trans. Am. Math. Soc. 318, 473–490 (1990).

    Google Scholar 

  11. Lamken E.R.: Existence results for generalized balanced tournament designs with block size 3. Des. Codes Cryptogr. 3, 33–61 (1992).

    Google Scholar 

  12. Lamken E.R.: Constructions for generalized balanced tournament designs. Discret. Math. 131, 127–151 (1994).

    Google Scholar 

  13. Rosa A., Vanstone S.A.: Starter–adder techniques for Kirkman squares and Kirkman cubes of small sides. Ars Combin. 14, 199–212 (1982).

    Google Scholar 

  14. Schellenberg P.J., Van Rees G.H.J., Vanstone S.A.: The existence of balanced tournament designs. Ars Combin. 3, 303–318 (1977).

    Google Scholar 

  15. Semakov N.V., Zinoviev V.A.: Equidistant \(q\)-ary codes with maximal distance and resolvable balanced incomplete block designs. Probl. Peredachi Inf. 4(2), 3–10 (1968). [Probl. Inf. Trans. (Engl. Transl.) 4(2), 1–7 (1968).]

  16. Semakov N.V., Zaitsev G.V., Zinoviev V.A.: A class of maximal equidistant codes. Probl. Peredachi Inf. 5(2), 84–87 (1969). [Probl. Inf. Trans. (Engl. Transl.) 5(2), 65–68 (1969).]

    Google Scholar 

  17. Shan X.: Near generalized balanced tournament designs with block sizes 4 and 5. Sci. China Ser. A. 50, 1382–1388 (2007).

    Google Scholar 

  18. Stanton R.G., Mullin R.C.: Construction of room squares. Ann. Math. Stat. 39, 1540–1548 (1968).

    Google Scholar 

  19. Yan J., Wang C.: The existence of FGDRP\((3, g^u)\)’s. Electron. J. Comb. 16, 1 (2009).

    Google Scholar 

  20. Yan J., Wang C.: The existence of near generalized balanced tournament designs. Electron. J. Comb. 19, 2 (2012).

    Google Scholar 

  21. Yan J., Yin J.: Constructions of optimal GDRP\((n,\lambda ; v)\)’s of type \(\lambda ^1\mu ^{m-1}\). Discret. Appl. Math. 156, 2666–2678 (2008).

  22. Yan J., Yin J.: A class of optimal constant composition codes from GDRPs. Des. Codes Cryptogr. 50, 61–76 (2009).

    Google Scholar 

  23. Yin J., Yan J., Wang C.: Generalized balanced tournament designs and related codes. Des. Codes Cryptogr. 46, 211–230 (2008).

    Google Scholar 

Download references

Acknowledgments

Research supported by the National Natural Science Foundation of China under Grant No. 11271280.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jianxing Yin.

Additional information

Communicated by V. A. Zinoviev.

Appendix: The required starter–adder pairs in the Proof of Lemma 3.3

Appendix: The required starter–adder pairs in the Proof of Lemma 3.3

\(u\)

  \(S\)

\(A\)

\(u\)

  \(S\)

\(A\)

\(u\)

  \(S\)

\(A\)

 

\(\{ 1_3, 4_4, 2_0, 3_3, \infty _0\}\)

\(0_0\)

 

\(\{ 1_2, 2_4, 3_0, 5_1,\infty _0 \}\)

\(2_0\)

 

\(\{ 2_0, 1_2, 3_3, 6_0, \infty _0\}\)

\(2_0\)

 

\(\{ 2_3, 1_4, 0_0, 4_0, \infty _1\}\)

\(2_0\)

 

\(\{ 0_4, 3_2, 4_2, 5_2,\infty _1 \}\)

\(4_0\)

 

\(\{ 1_4, 3_4, 4_4, 5_1, \infty _1\}\)

\(0_0\)

 

\(\{ 1_0, 0_3, 3_2, 4_2, \infty _2\}\)

\(3_0\)

 

\(\{ 3_4, 1_3, 4_0, 0_0,\infty _2 \}\)

\(1_0\)

 

\(\{ 2_2, 5_4, 4_3, 6_3, \infty _2\}\)

\(4_0\)

5

\(\{ 1_1, 3_4, 2_1, 0_1, \infty _3\}\)

\(1_0\)

6

\(\{ 1_1, 2_3, 5_4, 0_3,\infty _3 \}\)

\(3_0\)

7

\(\{ 2_1, 5_0, 0_3, 3_1, \infty _3\}\)

\(6_0\)

 

\(\{ 2_2, 3_1, 4_3, 0_2, \infty _4\}\)

\(4_0\)

 

\(\{ 2_2, 3_1, 4_1, 0_1,\infty _4 \}\)

\(5_0\)

 

\(\{ 1_1, 4_2, 6_2, 0_2, \infty _4\}\)

\(5_0\)

 

\(\{ 1_2, 3_0, 2_4, 4_1, 0_4 \}\)

\(R\)

 

\(\{ 2_0, 1_4, 3_3, 5_0, 4_3 \}\)

\(0_0\)

 

\(\{ 2_4, 4_0, 1_3, 5_2, 0_1, \}\)

\(1_0\)

 

\(\{ 2_ 2, 3_ 4, 7_ 3, 0_ 0, \infty _0\}\)

\(6_0\)

 

\(\{ 2_1, 4_4, 1_0, 5_3, 0_2 \}\)

\(R\)

 

\(\{ 1_0, 6_4, 2_3, 3_2, 0_4 \}\)

\(3_0\)

 

\(\{ 3_ 3, 6_ 3, 7_ 4, 9_ 3, \infty _1\}\)

\(8_0\)

 

\(\{ 4_ 1, 5_ 4, 6_ 3, 8_ 3, \infty _0\}\)

\(7_0\)

 

\(\{ 3_0, 0_0, 5_3, 4_1, 6_1 \}\)

\(R\)

 

\(\{ 1_ 4, 5_ 3, 7_ 1, 8_ 1, \infty _2\}\)

\(4_0\)

 

\(\{ 1_ 1, 2_ 3, 4_ 0, 7_ 1, \infty _1\}\)

\(3_0\)

 

\(\{ 2_ 3, 3_ 4, 5_ 1, 6_ 1, \infty _0\}\)

\(4_0\)

 

\(\{ 1_ 3, 4_ 4, 5_ 1, 9_ 4, \infty _3\}\)

\(3_0\)

 

\(\{ 2_ 4, 5_ 2, 7_ 0, 0_ 3, \infty _2\}\)

\(1_0\)

 

\(\{ 1_ 3, 6_ 4, 7_ 0, 0_ 3, \infty _1\}\)

\(3_0\)

 

\(\{ 2_ 3, 7_ 2, 8_ 2, 0_ 2, \infty _4\}\)

\(9_0\)

 

\(\{ 3_ 3, 4_ 3, 5_ 3, 0_ 2, \infty _3\}\)

\(2_0\)

 

\(\{ 1_ 1, 2_ 0, 5_ 4, 0_ 4, \infty _2\}\)

\(5_0\)

10

\(\{ 1_ 2, 3_ 0, 4_ 3, 8_ 0, 9_ 0 \}\)

\(5_0\)

9

\(\{ 1_ 3, 3_ 2, 8_ 4, 0_ 0, \infty _4\}\)

\(4_0\)

8

\(\{ 2_ 1, 4_ 1, 6_ 0, 7_ 4, \infty _3\}\)

\(0_0\)

 

\(\{ 2_ 1, 4_ 1, 5_ 2, 6_ 0, 9_ 2 \}\)

\(7_0\)

 

\(\{ 2_ 1, 3_ 0, 7_ 4, 8_ 2, 0_ 4 \}\)

\(8_0\)

 

\(\{ 3_ 2, 4_ 3, 6_ 3, 0_ 1, \infty _4\}\)

\(7_0\)

 

\(\{ 4_ 2, 5_ 0, 6_ 1, 7_ 0, 8_ 4 \}\)

\(0_0\)

 

\(\{ 3_ 1, 6_ 4, 7_ 3, 8_ 0, 0_ 1 \}\)

\(5_0\)

 

\(\{ 1_ 4, 2_ 2, 3_ 0, 4_ 4, 7_ 1 \}\)

\(2_0\)

 

\(\{ 1_ 1, 2_ 0, 3_ 1, 6_ 4, 0_ 4 \}\)

\(2_0\)

 

\(\{ 1_ 0, 2_ 0, 5_ 1, 6_ 0, 8_ 1 \}\)

\(0_0\)

 

\(\{ 2_ 4, 3_ 1, 4_ 2, 5_ 0, 0_ 2 \}\)

\(1_0\)

 

\(\{ 1_ 0, 2_ 4, 4_ 0, 6_ 2, 0_ 3 \}\)

\(1_0\)

 

\(\{ 1_ 4, 2_ 2, 4_ 2, 6_ 1, 7_ 2 \}\)

\(6_0\)

 

\(\{ 1_ 2, 3_ 3, 5_ 3, 6_ 2, 7_ 2 \}\)

\(6_0\)

 

\(\{ 3_ 2, 5_ 4, 8_ 3, 9_ 1, 0_ 1 \}\)

\(R\)

 

\(\{ 1_ 2, 3_ 4, 4_ 4, 5_ 0, 6_ 2 \}\)

\(R\)

 

\(\{ 1_ 0, 4_ 0, 5_ 2, 7_ 3, 0_ 0 \}\)

\(R\)

 

\(\{ 1_ 1, 2_ 2, 8_ 1, 0_ 1, \infty _0\}\)

\(8_0\)

 

\(\{ 4_ 3, 5_ 4, 7_ 1, 11_ 4, \infty _0\}\)

\(3_0\)

 

\(\{ 2_ 1, 7_ 4, 11_ 3, 0_ 1, \infty _0\}\)

\(8_0\)

 

\(\{ 3_ 1, 4_ 1, 5_ 4, 10_ 4, \infty _1\}\)

\(4_0\)

 

\(\{ 6_ 0, 8_ 1, 9_ 4, 10_ 1, \infty _1\}\)

\(9_0\)

 

\(\{ 5_ 1, 9_ 1, 10_ 2, 11_ 0, \infty _1\}\)

\(7_0\)

 

\(\{ 3_ 0, 6_ 0, 8_ 2, 10_ 1, \infty _2\}\)

\(9_0\)

 

\(\{ 6_ 1, 9_ 2, 10_ 3, 0_ 1, \infty _2\}\)

\(7_0\)

 

\(\{ 2_ 2, 6_ 1, 8_ 0, 0_ 0, \infty _2\}\)

\(12_0\)

 

\(\{ 1_ 2, 4_ 3, 6_ 3, 8_ 0, \infty _3\}\)

\(6_0\)

 

\(\{ 2_ 3, 3_ 4, 4_ 2, 0_ 2, \infty _3\}\)

\(6_0\)

 

\(\{ 3_ 1, 7_ 1, 11_ 2, 12_ 3, \infty _3\}\)

\(5_0\)

 

\(\{ 3_ 2, 4_ 4, 9_ 1, 0_ 0, \infty _4\}\)

\(3_0\)

 

\(\{ 1_ 4, 4_ 1, 10_ 4, 11_ 1, \infty _4\}\)

\(10_0\)

 

\(\{ 1_ 2, 5_ 2, 10_ 0, 0_ 3, \infty _4\}\)

\(9_0\)

11

\(\{ 1_ 3, 4_ 2, 5_ 0, 7_ 0, 9_ 0 \}\)

\(1_0\)

12

\(\{ 3_ 0, 6_ 2, 8_ 3, 10_ 0, 0_ 0 \}\)

\(1_0\)

13

\(\{ 1_ 4, 6_ 3, 8_ 4, 9_ 4, 12_ 0 \}\)

\(10_0\)

 

\(\{ 6_ 4, 7_ 2, 8_ 3, 9_ 3, 10_ 2 \}\)

\(7_0\)

 

\(\{ 1_ 3, 2_ 2, 4_ 0, 9_ 1, 11_ 3 \}\)

\(2_0\)

 

\(\{ 2_ 4, 5_ 4, 7_ 2, 9_ 0, 12_ 2 \}\)

\(2_0\)

 

\(\{ 2_ 3, 6_ 2, 7_ 1, 10_ 0, 0_ 2 \}\)

\(2_0\)

 

\(\{ 2_ 0, 5_ 1, 6_ 4, 7_ 0, 11_ 0 \}\)

\(11_0\)

 

\(\{ 1_ 0, 6_ 2, 8_ 2, 9_ 3, 0_ 2 \}\)

\(1_0\)

 

\(\{ 2_ 0, 3_ 4, 5_ 2, 10_ 3, 0_ 3 \}\)

\(10_0\)

 

\(\{ 1_ 2, 3_ 2, 5_ 3, 8_ 4, 0_ 3 \}\)

\(5_0\)

 

\(\{ 3_ 4, 4_ 1, 5_ 0, 6_ 0, 9_ 2 \}\)

\(6_0\)

 

\(\{ 3_ 3, 4_ 0, 5_ 1, 7_ 3, 9_ 4 \}\)

\(0_0\)

 

\(\{ 1_ 0, 5_ 0, 7_ 4, 10_ 2, 11_ 2 \}\)

\(4_0\)

 

\(\{ 3_ 2, 4_ 4, 5_ 3, 8_ 1, 11_ 4 \}\)

\(3_0\)

 

\(\{ 1_ 0, 2_ 4, 5_ 3, 8_ 4, 0_ 4 \}\)

\(5_0\)

 

\(\{ 2_ 1, 3_ 1, 7_ 3, 8_ 0, 9_ 0 \}\)

\(0_0\)

 

\(\{ 4_ 3, 6_ 4, 7_ 0, 10_ 4, 0_ 4 \}\)

\(11_0\)

 

\(\{ 1_ 4, 2_ 1, 6_ 1, 7_ 4, 9_ 2 \}\)

\(R\)

 

\(\{ 2_ 4, 3_ 3, 6_ 3, 7_ 2, 8_ 2 \}\)

\(8_0\)

 

\(\{ 2_ 3, 3_ 0, 4_ 0, 10_ 1, 12_ 4 \}\)

\(0_0\)

 

\(\{ 1_ 1, 2_ 0, 3_ 0, 13_ 3, \infty _0\}\)

\(2_0\)

 

\(\{ 1_ 1, 4_ 4, 5_ 2, 9_ 3, 0_ 4 \}\)

\(R\)

 

\(\{ 1_ 3, 2_ 0, 3_ 3, 4_ 2, 12_ 1 \}\)

\(4_0\)

 

\(\{ 2_ 2, 3_ 2, 10_ 2, 0_ 1, \infty _1\}\)

\(9_0\)

 

\(\{ 3_ 1, 4_ 2, 5_ 3, 7_ 0, \infty _0\}\)

\(13_0\)

 

\(\{ 1_ 1, 7_ 3, 8_ 3, 10_ 3, 11_ 1 \}\)

\(R\)

 

\(\{ 5_ 0, 13_ 1, 15_ 3, 0_ 4, \infty _2\}\)

\(6_0\)

 

\(\{ 2_ 0, 7_ 3, 1_ 0, 3_ 4, \infty _1\}\)

\(5_0\)

 

\(\{4_ 1, 9_ 2, 10_ 2, 11_ 0, \infty _0\}\)

\(6_0\)

 

\(\{ 3_ 4, 6_ 1, 9_ 3, 10_ 0, \infty _3\}\)

\(4_0\)

 

\(\{11_ 0, 0_ 4, 2_ 2, 14_ 2, \infty _2\}\)

\(11_0\)

 

\(\{1_ 4, 2_ 3, 6_ 1, 10_ 1, \infty _1\}\)

\(7_0\)

 

\(\{ 6_ 2, 8_ 4, 9_ 0, 10_ 4, \infty _4\}\)

\(15_0\)

 

\(\{10_ 0, 4_ 4, 3_ 0, 5_ 1, \infty _3\}\)

\(6_0\)

 

\(\{0_ 0, 13_ 4, 9_ 3, 11_ 2, \infty _2\}\)

\(12_0\)

 

\(\{ 5_ 2, 11_ 0, 12_ 1, 13_ 0, 0_ 3 \}\)

\(10_0\)

 

\(\{10_ 1, 9_ 2, 7_ 1, 1_ 1, \infty _4\}\)

\(10_0\)

 

\(\{2_ 4, 4_ 2, 9_ 1, 10_ 4, \infty _3\}\)

\(11_0\)

 

\(\{ 1_ 0, 2_ 3, 8_ 1, 10_ 3, 14_ 1 \}\)

\(0_0\)

 

\(\{ 9_ 0, 13_ 3, 12_ 1, 3_ 2, 2_ 1 \}\)

\(1_0\)

 

\(\{1_ 3, 5_ 0, 6_ 3, 13_ 2, \infty _4\}\)

\(10_0\)

 

\(\{ 1_ 2, 4_ 4, 11_ 3, 12_ 2, 0_ 2 \}\)

\(13_0\)

 

\(\{ 2_ 3, 13_ 1, 5_ 2, 6_ 0, 11_ 4 \}\)

\(7_0\)

 

\(\{5_ 1, 2_ 0, 12_ 1, 7_ 0, 8_ 0 \}\)

\(1_0\)

 

\(\{ 1_ 4, 5_ 3, 7_ 4, 8_ 2, 11_ 4 \}\)

\(1_0\)

 

\(\{ 8_ 3, 10_ 2, 6_ 2, 13_ 0, 12_ 0 \}\)

\(12_0\)

 

\(\{6_ 4, 11_ 4, 1_ 1, 9_ 4, 7_ 1 \}\)

\(9_0\)

16

\(\{ 2_ 4, 5_ 4, 7_ 3, 9_ 2, 14_ 0 \}\)

\(12_0\)

15

\(\{ 1_ 4, 13_ 4, 8_ 2, 6_ 4, 7_ 4 \}\)

\(0_0\)

14

\(\{4_ 3, 7_ 2, 12_ 4, 3_ 1, 0_ 4 \}\)

\(5_0\)

 

\(\{ 6_ 0, 10_ 1, 12_ 4, 14_ 2, 15_ 4 \}\)

\(14_0\)

 

\(\{ 9_ 3, 12_ 2, 6_ 3, 14_ 4, 10_ 3 \}\)

\(2_0\)

 

\(\{13_ 0, 1_ 2, 6_ 0, 0_ 1, 3_ 2 \}\)

\(4_0\)

 

\(\{ 2_ 1, 8_ 3, 9_ 1, 13_ 4, 15_ 2 \}\)

\(5_0\)

 

\(\{13_ 2, 0_ 2, 1_ 3, 8_ 0, 4_ 0 \}\)

\(4_0\)

 

\(\{8_ 3, 2_ 2, 11_ 3, 3_ 4, 0_ 2 \}\)

\(2_0\)

 

\(\{ 4_ 0, 8_ 0, 11_ 1, 13_ 2, 15_ 0 \}\)

\(7_0\)

 

\(\{ 2_ 4, 7_ 2, 0_ 0, 8_ 1, 11_ 3 \}\)

\(14_0\)

 

\(\{2_ 1, 8_ 4, 12_ 3, 1_ 0, 4_ 4 \}\)

\(3_0\)

 

\(\{ 3_ 3, 4_ 3, 6_ 4, 9_ 4, 14_ 3 \}\)

\(3_0\)

 

\(\{14_ 3, 11_ 2, 3_ 3, 8_ 4, 0_ 1 \}\)

\(3_0\)

 

\(\{4_ 0, 3_ 0, 5_ 4, 0_ 3, 12_ 2 \}\)

\(8_0\)

 

\(\{ 1_ 3, 4_ 1, 6_ 3, 7_ 0, 0_ 0 \}\)

\(11_0\)

 

\(\{ 5_ 4, 12_ 4, 10_ 4, 14_ 0, 0_ 3 \}\)

\(9_0\)

 

\(\{7_ 3, 8_ 2, 5_ 2, 3_ 3, 13_ 3 \}\)

\(13_0\)

 

\(\{ 3_ 1, 5_ 1, 7_ 1, 11_ 2, 12_ 3 \}\)

\(8_0\)

 

\(\{11_ 1, 4_ 3, 9_ 1, 1_ 2, 5_ 0 \}\)

\(8_0\)

 

\(\{5_ 3, 8_ 1, 10_ 3, 11_ 1, 12_ 0 \}\)

\(0_0\)

 

\(\{ 4_ 2, 7_ 2, 12_ 0, 14_ 4, 15_ 1 \}\)

\(R\)

 

\(\{ 9_ 4, 14_ 1, 12_ 3, 4_ 1, 6_ 1 \}\)

\(R\)

 

\(\{7_ 4, 6_ 2, 10_ 0, 9_ 0, 13_ 1 \}\)

\(R\)

\(u\)

  \(S\)

\(A\)

  \(S\)

\(A\)

  \(S\)

\(A\)

 

\(\{ 3_ 3, 13_ 3, 0_ 4, 15_ 1, \infty _0\}\)

\(9_0\)

\(\{ 3_ 0, 5_ 2, 10_ 4, 2_ 3, 11_ 2 \}\)

\(2_0\)

\(\{15_ 0, 4_ 3, 6_ 0, 0_ 1, 1_ 2 \}\)

\(3_0\)

 

\(\{ 6_ 3, 11_ 0, 13_ 2, 9_ 0, \infty _1\}\)

\(5_0\)

\(\{ 1_ 1, 16_ 1, 2_ 4, 11_ 3, 4_ 0 \}\)

\(7_0\)

\(\{14_ 4, 13_ 4, 5_ 3, 11_ 4, 15_ 3 \}\)

\(14_0\)

17

\(\{10_ 0, 9_ 1, 3_ 4, 8_ 4, \infty _2\}\)

\(13_0\)

\(\{11_ 1, 14_ 1, 2_ 0, 3_ 1, 16_ 4 \}\)

\(16_0\)

\(\{ 5_ 0, 13_ 0, 16_ 3, 9_ 3, 8_ 1 \}\)

\(6_0\)

 

\(\{ 6_ 1, 16_ 0, 10_ 1, 0_ 3, \infty _3\}\)

\(10_0\)

\(\{ 8_ 2, 12_ 1, 15_ 4, 16_ 2, 4_ 2 \}\)

\(15_0\)

\(\{ 2_ 2, 4_ 4, 7_ 2, 3_ 2, 13_ 1 \}\)

\(12_0\)

 

\(\{12_ 2, 1_ 3, 7_ 1, 14_ 0, \infty _4\}\)

\(8_0\)

\(\{ 9_ 4, 0_ 2, 8_ 0, 12_ 4, 7_ 0 \}\)

\(4_0\)

\(\{ 4_ 1, 1_ 0, 9_ 2, 10_ 3, 14_ 2 \}\)

\(11_0\)

 

\(\{ 0_ 0, 2_ 1, 5_ 1, 6_ 2, 12_ 3 \}\)

\(1_0\)

\(\{ 8_ 3, 10_ 2, 14_ 3, 7_ 3, 6_ 4 \}\)

\(0_0\)

\(\{ 7_ 4, 15_ 2, 5_ 4, 1_ 4, 12_ 0 \}\)

\(R\)

 

\(\{ 3_ 2, 7_ 0, 22_ 0, 26_ 2, \infty _0\}\)

\(5_0\)

\(\{ 9_ 3, 3_ 1, 26_ 4, 8_ 0, 23_ 0 \}\)

\(12_0\)

\(\{19_ 0, 12_ 2, 11_ 3, 10_ 4, 27_ 1 \}\)

\(14_0\)

 

\(\{21_ 1, 4_ 3, 14_ 2, 11_ 2, \infty _1\}\)

\(3_0\)

\(\{20_ 3, 26_ 1, 3_ 4, 21_ 0, 6_ 0 \}\)

\(19_0\)

\(\{25_ 0, 28_ 2, 16_ 3, 4_ 4, 5_ 1 \}\)

\(10_0\)

 

\(\{20_ 1, 19_ 3, 23_ 2, 16_ 2, \infty _2\}\)

\(4_0\)

\(\{ 8_ 3, 22_ 1, 7_ 4, 20_ 0, 14_ 0 \}\)

\(13_0\)

\(\{20_ 4, 2_ 4, 5_ 2, 10_ 2, 18_ 0 \}\)

\(28_0\)

 

\(\{ 8_ 1, 25_ 3, 15_ 2, 18_ 2, \infty _3\}\)

\(24_0\)

\(\{ 1_ 0, 11_ 4, 13_ 1, 14_ 1, 26_ 3 \}\)

\(17_0\)

\(\{ 8_ 4, 24_ 4, 2_ 2, 4_ 2, 13_ 0 \}\)

\(9_0\)

29

\(\{ 9_ 1, 10_ 3, 6_ 2, 13_ 2, \infty _4\}\)

\(6_0\)

\(\{12_ 0, 16_ 4, 11_ 1, 23_ 1, 22_ 3 \}\)

\(25_0\)

\(\{ 9_ 4, 27_ 4, 24_ 2, 19_ 2, 11_ 0 \}\)

\(15_0\)

 

\(\{12_ 1, 1_ 3, 17_ 4, 20_ 2, 25_ 1 \}\)

\(22_0\)

\(\{28_ 0, 18_ 4, 16_ 1, 15_ 1, 3_ 3 \}\)

\(16_0\)

\(\{21_ 4, 5_ 4, 27_ 2, 25_ 2, 16_ 0 \}\)

\(7_0\)

 

\(\{28_ 1, 12_ 3, 1_ 4, 8_ 2, 10_ 1 \}\)

\(26_0\)

\(\{17_ 0, 13_ 4, 18_ 1, 6_ 1, 7_ 3 \}\)

\(8_0\)

\(\{ 0_ 2, 2_ 3, 23_ 3, 6_ 3, 27_ 3 \}\)

\(1_0\)

 

\(\{17_ 1, 28_ 3, 12_ 4, 9_ 2, 4_ 1 \}\)

\(23_0\)

\(\{10_ 0, 17_ 2, 18_ 3, 19_ 4, 2_ 1 \}\)

\(20_0\)

\(\{ 0_ 3, 6_ 4, 14_ 4, 15_ 4, 23_ 4 \}\)

\(2_0\)

 

\(\{ 1_ 1, 17_ 3, 28_ 4, 21_ 2, 19_ 1 \}\)

\(0_0\)

\(\{ 4_ 0, 1_ 2, 13_ 3, 25_ 4, 24_ 1 \}\)

\(27_0\)

\(\{ 0_ 1, 3_ 0, 7_ 2, 22_ 2, 26_ 0 \}\)

\(11_0\)

 

\(\{ 0_ 4, 24_ 3, 5_ 3, 15_ 3, 14_ 3 \}\)

\(21_0\)

\(\{ 0_ 0, 2_ 0, 24_ 0, 27_ 0, 5_ 0 \}\)

\(18_0\)

\(\{21_ 3, 7_ 1, 22_ 4, 9_ 0, 15_ 0 \}\)

\(R\)

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Dai, P., Wang, J. & Yin, J. Two series of equitable symbol weight codes meeting the Plotkin bound. Des. Codes Cryptogr. 74, 15–29 (2015). https://doi.org/10.1007/s10623-013-9846-z

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  • DOI: https://doi.org/10.1007/s10623-013-9846-z

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