广西师范大学学报（自然科学版） ›› 2021, Vol. 39 ›› Issue (6): 119-129.doi: 10.16088/j.issn.1001-6600.2021040801

• 研究论文 • 上一篇    下一篇

广义Howell设计GHD(n+5,3n)的存在性

姚金洋, 胡颖, 王金华*   

  1. 南通大学 理学院, 江苏 南通 226007
  • 收稿日期:2021-04-08 修回日期:2021-05-18 出版日期:2021-11-25 发布日期:2021-12-08
  • 通讯作者: 王金华(1963—), 男, 江苏如东人, 南通大学教授, 博士。E-mail: jhwang@ntu.edu.cn
  • 基金资助:
    国家自然科学基金(11371207)

Existence of Generalized Howell Designs GHD(n+5,3n)s

YAO Jinyang, HU Ying, WANG Jinhua*   

  1. School of Sciences, Nantong University, Nantong Jiangsu 226007, China
  • Received:2021-04-08 Revised:2021-05-18 Online:2021-11-25 Published:2021-12-08

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摘要： 广义Howell设计是一类双可分解设计,与置换表、多层常重码有密切联系。本文利用可迁和不可迁starter-adder直接构造方法和广义Howell标架递推工具,给出广义Howell设计新的构造,除了53个可能例外值,解决了每行和每列恰好有5个空单元格的广义Howell设计GHD(n+5,3n)的存在性问题。利用广义Howell设计和多层常重码之间的关系,得到相应最优多层常重码MCWC(3,3n;1,n+5;1,n+5;8)的存在性。

关键词: 广义Howell设计, 多层常重码, 广义Howell标架, starter-adder

Abstract: Generalized Howell design is a kind of double resolvable designs, which are closely related to permutation arrays and multiply constant-weight codes. By making full use of the direct construction method of transitive starter-adder, intransitive starter-adder and generalized Howell frames as recursive tool, some new constructions for generalized Howell designs are given in this paper. The problem of existence of the generalized Howell design GHD(n+5,3n)s with exactly 5 empty cells in each row and column is solved with 53 possible exceptions. Then, the existence of the corresponding optimal multiply constant-weight codes MCWC(3,3n;1,n+5;1,n+5;8) is given by using the relationship between the generalized Howell designs and the multiply constant-weight codes.

Key words: generalized Howell design, multiply constant-weight code, generalized Howell frame, starter-adder

中图分类号: 

  • O157.2
[1] DEZA M, VANSTONE S A. Bounds for permutation arrays[J]. Journal of Statistical Planning and Inference, 1978, 2(2): 197-209.
[2] ETZION T. Optimal doubly constant weight codes[J]. Journal of Combinatorial Designs, 2008, 16(2): 137-151.
[3] CHEE Y M, KIAH H M, ZHANG H, et al. Constructions of optimal and near optimal multiply constant-weight codes[J]. IEEE Transactions on Information Theory, 2017, 63(6): 3621-3629.
[4] WANG C Y, CHANG Y X, FENG T. Optimal multiply constant-weight codes from generalized Howell designs[J]. Graphs and Combinatorics, 2019, 35(3): 611-632.
[5] WANG C Y, CHANG Y X, FENG T. The asymptotic existence of frames with a pair of orthogonal frame resolutions[J]. Science China Mathematics, 2019, 62(9):1839-1850.
[6] COLBOURN C J, DINITZ J H. The CRC Handbook of combinatorial designs[M]. 2nd ed. Boca Raton, FL:Chapman & Hall/CRC Press, 2007.
[7] MULLIN R C, WALLIS W D. The existence of Room squares[J]. Aequationes Mathematicae, 1975, 13(1/2): 1-7.
[8] STINSON D R. The existence of Howell designs of odd side[J]. Journal of Combinatorial Theory (Series A), 1982, 32(1): 53-65.
[9] ANDERSON B A, SCHELLENBERG P J, STINSON D R: The existence of Howell designs of even side[J]. Journal of Combinatorial Theory (Series A), 1984, 36(1): 23-55.
[10] ABEL R J R, BAILEY R F, BURGESS A C, et al. On generalized Howell designs with block size three[J]. Designs, Codes and Cryptography, 2016, 81(2): 365-391.
[11] WANG C Y, DU B L. Existence of generalized Howell designs of side n+1 and order 3n[J].Utilitas Mathematica, 2009, 80: 143-159.
[12] 王长远,冯弢. 每行(列)含2个空位的广义Howell设计[J]. 北京交通大学学报,2016, 40(3):120-128.
[13] COLBOURN C J, LAMKEN E R, LING A C H, et al. The existence of Kirkman squares-doubly resolvable (v,3,1)-BIBDs[J]. Designs, Codes and Cryptography, 2002, 26(1/2/3): 169-196.
[14] MATHON R, VANSTONE S A. On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays[J]. Discrete Mathematics, 1980, 30(2): 157-172.
[15] ABEL R J R, LAMKEN E R, WANG J. A few more Kirkman squares and doubly near resolvable BIBDs with block size 3[J]. Discrete Mathematics, 2008, 308(7): 1102-1123.
[16] ABEL R J R, CHAN N, COLBOURN C J, et al. Doubly resolvable nearly Kirkman triple systems[J]. Journal of Combinatorial Designs, 2013, 21(8): 342-358.
[17] KOTZIG A, ROSA A. Nearly Kirkman systems[J]. Congressus Numerantium, 1974, 10: 607-614.
[18] SMITH P. A doubly divisible nearly Kirkman system[J]. Discrete Mathematics, 1977, 18(1): 93-96.
[19] COLBOURN C J , HORSLEY D, WANG C. Colouring triples in every way: A conjecture[J]. Quaderni di Matematica, 2012, 28: 257-286.
[20] DU J, ABEL R J R, WANG J H. Some new resolvable GDDs with k=4 and doubly resolvable GDDs with k=3[J]. Discrete Mathematics, 2015, 338(11): 2015-2118.
[21] ABEL R J R, BENNETT F E, GE G. The existence of four HMOLS with equal sized holes[J]. Designs, Codes and Cryptography, 2002, 26: 7-31.
[22] BENNETT F E, COLBOURN C J, ZHU L. Existence of three HMOLS of types hn and 2n31 [J]. Discrete Mathematics, 1996, 160(1/3): 49-65.
[23] ABEL R J R. Existence of five MOLS of orders 18 and 60[J]. Journal of Combinatorial Designs, 2015, 23(4): 135-139.
[24] TODOROV D T. Four mutually orthogonal Latin squares of order 14[J]. Journal of Combinatorial Designs, 2012, 20(8): 363-367.
[25] CHERIF Z, DANGER J L, GUILLEY S, et al. Multiply constant weight codes[C]//2013 IEEE International Symposium on Information Theory, Piscataway NJ: IEEE Press, 2013: 306-310.
[26] CHEE Y M, CHERIF Z, DANGER J L, et al. Multiply constant-weight codes and the reliability of loop physically unclonable functions[J]. IEEE Transactions on Information Theory, 2014, 60(11): 7026-7034.
[27] WANG X, WEI H J, SHANGGUAN C, et al. New bounds and constructions for multiply constant-weight codes[J]. IEEE Transactions on Information Theory, 2016, 62(11): 6315-6327.
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