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Designs, Codes and Cryptography

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21-10-2022

Guest editorial: On coding theory and combinatorics—in memory of Vera Pless

Authors: W. Cary Huffman, Jon-Lark Kim, Patrick Solé

Published in: Designs, Codes and Cryptography | Issue 11/2022

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This special issue of Designs, Codes and Cryptography is dedicated to Vera Stepen Pless, our beloved collaborator, advisor, and friend. Accounts of her biography and bibliography have appeared in several places [4, 20, 28, 29, 67]. The topics of the articles that comprise the issue reflect the themes and trends of Vera’s long and fertile research career. We consider her career by examining her impact on the following areas of coding theory:

self-orthogonal and self-dual codes,

formally self-dual codes,

identities on the weight distribution,

covering radius,

families of linear codes,

additive codes,

codes and block designs,

codes over rings,

decoding, and

10.

cryptography

Also within each of Sects. 1 through 10 (except Sect. 4) we include brief synopses of the articles in this issue that pertain to the topic of the section; additional papers in this issue are then listed in Sect. 11. We conclude with a description of Vera’s books and then present a concluding summary and tribute to Vera Pless. …

next article Weight distribution of double cyclic codes over Galois rings

Assmus E.F. Jr., Pless V.S.: On the covering radius of extremal self-dual codes. IEEE Trans. Inform. Theory 29, 359–363 (1983).MathSciNetMATHCrossRef

Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18, 11–28 (1999) Designs and codes—a memorial tribute to Ed Assmus.MathSciNetMATHCrossRef

Beissinger J., Pless V.S.: The Cryptoclub: Using Mathematics to Make and Break Secret Codes. CRC Press, Boca Raton (2006).

Biographies of Women Mathematicians. https://mathwomen.agnesscott.org/women/pless.htm.

Bouyuklieva S.: Self-dual codes. In: Huffman W.C., Kim J.-L., Solé P. (eds.) Concise Encyclopedia of Coding Theory, Chapter 4, pp. 79–96. CRC Press, Boca Raton (2021).

Brualdi R.A., Pless V.S.: On the length of codes with a given covering radius. In: Coding Theory and Design Theory, Part I, volume 20 of IMA Vol. Math. Appl., pp. 9–15. Springer, New York (1990).

Brualdi R.A., Pless V.S.: Subcodes of Hamming codes. In Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989), vol. 70, pp. 153–158 (1990).

Brualdi R.A., Pless V.S.: Orphans of the first order Reed–Muller codes. IEEE Trans. Inform. Theory 36, 399–401 (1990).MathSciNetMATHCrossRef

Brualdi R.A., Pless V.S.: On the covering radius of a code and its subcodes. Discret. Math. 83, 189–199 (1990).MathSciNetMATHCrossRef

10.

Brualdi R.A., Pless V.S.: Greedy codes. J. Comb. Theory Ser. A 64, 10–30 (1993).MathSciNetMATHCrossRef

11.

Brualdi R.A., Pless V.S., Wilson R.M.: Short codes with a given covering radius. IEEE Trans. Inform. Theory 35, 99–109 (1989).MathSciNetMATHCrossRef

12.

Brualdi R.A., Cai N., Pless V.S.: Orphan structure of the first order Reed–Muller codes. Discret. Math. 102, 239–247 (1992).MathSciNetMATHCrossRef

13.

Brualdi R.A., Litsyn S., Pless V.S.: Covering Radius. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, Vol. I, II, Chapter 8, pp. 755–826. North-Holland, Amsterdam (1998).

14.

Conway J.H., Pless V.S.: On the enumeration of self-dual codes. J. Comb. Theory Ser. A 28, 26–53 (1980).MathSciNetMATHCrossRef

15.

Conway J.H., Pless V.S.: On primes dividing the group order of a doubly-even \((72,36,16)\) code and the group order of a quaternary \((24,12,10)\) code. Discret. Math. 38, 143–156 (1982).MATHCrossRef

16.

Conway J.H., Pless V.S., Sloane N.J.A.: Self-dual codes over GF(3) and GF(4) of length not exceeding \(16\). IEEE Trans. Inform. Theory 25, 312–322 (1979).MathSciNetMATHCrossRef

17.

Conway J.H., Pless V.S., Sloane N.J.A.: The binary self-dual codes of length up to 32: a revised enumeration. J. Comb. Theory Ser. A 60, 183–195 (1992).MathSciNetMATHCrossRef

18.

Danielsen L.E.: Database of self-dual quantum codes. http://www.ii.uib.no/~larsed/vncorbits.

19.

Ding C., Pless V.S.: Cyclotomy and duadic codes of prime lengths. IEEE Trans. Inform. Theory 45, 453–466 (1999).MathSciNetMATHCrossRef

20.

Dougherty S.T., Matthews G., Wood J.: The life and work of Vera Stepen Pless. (2022).

21.

Fields J.E., Gaborit P., Leon J.S., Pless V.S.: All self-dual \(\mathbb{Z} _4\) codes of length \(15\) or less are known. IEEE Trans. Inform. Theory 44, 311–322 (1998).MathSciNetMATHCrossRef

22.

Fields J.E., Gaborit P., Huffman W.C., Pless V.S.: On the classification of extremal even formally self-dual codes. Des. Codes Cryptogr. 18, 125–148 (1999).MathSciNetMATHCrossRef

23.

Fields J.E., Gaborit P., Huffman W.C., Pless V.S.: On the classification of extremal even formally self-dual codes of lengths 20 and 22. Discret. Appl. Math. 111, 75–86 (2001).MathSciNetMATHCrossRef

24.

Gaborit P., Huffman W.C., Kim J.-L., Pless V.S.: On additive \(\rm GF(4)\) codes. In Codes and Association Schemes (Piscataway, NJ, 1999), volume 56 of DIMACS Ser. Discret. Math. Theoret. Comput. Sci. pp. 135–149. Amer. Math. Soc., Providence, RI (2001)

25.

Gaborit P., Pless V.S., Solé P., Atkin O.: Type II codes over \(\mathbb{F} _4\). Finite Fields Appl. 8, 171–183 (2002).MathSciNetMATHCrossRef

26.

Gaborit P., Kim J.-L., Pless V.S.: Decoding binary \(R(2,5)\) by hand. Discret. Math. 264, 55–73 (2003).MathSciNetMATHCrossRef

27.

Grassl M.: Code tables: Bounds on the parameters of various types of codes. http://www.codetables.de.

28.

Huffman W.C.: A tribute to Vera Pless (1931–2020). Finite Fields Appl. 65, 101675 (2020).MathSciNetMATHCrossRef

29.

Huffman W.C.: In Memorium: Vera Stepen Pless. https://www.itsoc.org/news-events/recent-news/in-memoriam-vera-stepen-pless.

30.

Huffman W.C., Pless V.S.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).MATHCrossRef

31.

Kennedy G., Pless V.S.: On designs and formally self-dual codes. Des. Codes Cryptogr. 4, 43–55 (1994).MathSciNetMATHCrossRef

32.

Kim J.-L., Pless V.S.: Decoding some doubly-even self-dual [32, 16, 8] codes by hand. In: Codes and Designs (Columbus, OH, 2000), vol. 10 of Ohio State Univ. Math. Res. Inst. Publ., pp. 165–178. de Gruyter, Berlin (2002)

33.

Kim J.-L., Pless V.S.: Designs in additive codes over \(\rm GF (4)\). Des. Codes Cryptogr. 30, 187–199 (2003).MathSciNetMATHCrossRef

34.

Kim J.-L., Pless V.S.: A note on formally self-dual even codes of length divisible by 8. Finite Fields Appl. 13, 224–229 (2007).MathSciNetMATHCrossRef

35.

Leon J.S., Pless V.S., Sloane N.J.A.: On ternary self-dual codes of length \(24\). IEEE Trans. Inform. Theory 27, 176–180 (1981).MathSciNetMATHCrossRef

36.

Leon J.S., Pless V.S., Sloane N.J.A.: Self-dual codes over \(\rm GF (5)\). J. Comb. Theory Ser. A 32, 178–194 (1982).MathSciNetMATHCrossRef

37.

Leon J.S., Masley J.M., Pless V.S.: Duadic codes. IEEE Trans. Inform. Theory 30, 709–714 (1984).MathSciNetMATHCrossRef

38.

MacWilliams F.J.: Combinatorial Problems of Elementary Abelian Groups. PhD thesis, Harvard University (1962)

39.

Mallows C.L., Pless V.S., Sloane N.J.A.: Self-dual codes over \(\rm GF (3)\). SIAM J. Appl. Math. 31, 649–666 (1976).MathSciNetMATHCrossRef

40.

Pless V.S.: Power moment identities on weight distributions in error correcting codes. Inform. Control 6, 147–152 (1963).MathSciNetMATHCrossRef

41.

Pless V. S.: The number of isotropic subspaces in a finite geometry. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 39, 418–421 (1965).MathSciNetMATH

42.

Pless V.S.: On the uniqueness of the Golay codes. J. Comb. Theory 5, 215–228 (1968).MathSciNetMATHCrossRef

43.

Pless V.S.: On a family of symmetry codes over \(\rm GF(3)\) and related new five-designs. In Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), pp. 323–326. Gordon and Breach, New York (1970).

44.

Pless V.S.: A classification of self-orthogonal codes over \(\rm GF (2)\). Discret. Math. 3, 209–246 (1972).MathSciNetMATHCrossRef

45.

Pless V.S.: CAMAC—Combinatorial and Algebraic Machine Aided Computation. In Proc. of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, FL, 1975), pp. 505–511. Congressus Numerantium, No,. XIV, (1975)

46.

Pless V.S.: The children of the \((32,\,16)\) doubly even codes. IEEE Trans. Inform. Theory 24, 738–746 (1978).MathSciNetMATHCrossRef

47.

Pless V.S.: \(23\) does not divide the order of the group of a \((72,36,16)\) doubly even code. IEEE Trans. Inform. Theory 28, 113–117 (1982).MathSciNetMATHCrossRef

48.

Pless V.S.: Decoding the Golay codes. IEEE Trans. Inform. Theory 32, 561–567 (1986).MathSciNetMATHCrossRef

49.

Pless V.S.: \(Q\)-codes. J. Comb. Theory Ser. A 43, 258–276 (1986).MATHCrossRef

50.

Pless V.S.: Duadic codes and generalizations. In: Camion P., Charpin P., Harari S. (eds.) Proc. International Symposium on Information Theory – EUROCODE 1992, volume 339 of CISM Courses and Lect., pages 3–15. Springer, Vienna (1993)

51.

Pless V.S.: Introduction to the Theory of Error-Correcting Codes, 3rd edn Wiley, New York (1998).MATHCrossRef

52.

Pless V.S., Huffman W.C. (eds.): Handbook of Coding Theory. II, vol. I. North-Holland, Amsterdam (1998).MATH

53.

Pless V.S., Pierce J.N.: Self-dual codes over \(\rm GF (q)\) satisfy a modified Varshamov–Gilbert bound. Inf. Control 23, 35–40 (1973).MathSciNetMATHCrossRef

54.

Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over \(\mathbb{Z} _4\). IEEE Trans. Inform. Theory 42, 1594–1600 (1996).MathSciNetMATHCrossRef

55.

Pless V.S., Rushanan J.J.: Triadic codes. Linear Algebra Appl. 98, 415–433 (1988).MathSciNetMATHCrossRef

56.

Pless V.S., Sloane N.J.A.: On the classification and enumeration of self-dual codes. J. Comb. Theory Ser. A 18, 313–335 (1975).MathSciNetMATHCrossRef

57.

Pless V.S., Thompson J.G.: \(17\) does not divide the order of the group of a \((72,36,16)\) doubly even code. IEEE Trans. Inform. Theory 28, 537–541 (1982).MathSciNetMATHCrossRef

58.

Pless V.S., Tonchev V.D.: Self-dual codes over \(\rm GF (7)\). IEEE Trans. Inform. Theory 33, 723–727 (1987).MathSciNetMATHCrossRef

59.

Pless V.S., Sloane N.J.A., Ward H.N.: Ternary codes of minimum weight \(6\) and the classification of the self-dual codes of length \(20\). IEEE Trans. Inform. Theory 26, 305–316 (1980).MathSciNetMATHCrossRef

60.

Pless V.S., Masley J.M., Leon J.S.: On weights in duadic codes. J. Comb. Theory Ser. A 44, 6–21 (1987).MathSciNetMATHCrossRef

61.

Pless V.S., Leon J.S., Fields J.E.: All \(\mathbb{Z} _4\) codes of type II and length \(16\) are known. J. Comb. Theory Ser. A 78, 32–50 (1997).MATHCrossRef

62.

Pless V.S., Solé P., Qian Z.: Cyclic self-dual \(\mathbb{Z} _4\)-codes. Finite Fields Appl. 3, 48–69 (1997).MathSciNetMATHCrossRef

63.

Rushanan J.J.: Generalized \(Q\)-Codes. PhD thesis, California Institute of Technology (1986)

64.

Simonis J.: The \([18,9,6]\) code is unique. Discret. Math. 106(107), 439–448 (1992).MathSciNetMATHCrossRef

65.

Sloane N.J.A.: Is there a \((72,36)\)\(d=16\) self-dual code? IEEE Trans. Inform. Theory 19, 251 (1973).MathSciNetMATHCrossRef

66.

Smid M.H.M.: Duadic codes. IEEE Trans. Inform. Theory 33, 432–433 (1987).MathSciNetMATHCrossRef

67.

Vera Pless. https://mathshistory.st-andrews.ac.uk/Biographies/Pless/.

Title: Guest editorial: On coding theory and combinatorics—in memory of Vera Pless
Authors: W. Cary Huffman
Jon-Lark Kim
Patrick Solé
Publication date: 21-10-2022
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 11/2022
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01126-z

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