Symplectic semifield planes and linear codes
Authors:
William M. Kantor and Michael E. Williams
Journal:
Trans. Amer. Math. Soc. 356 (2004), 895938
MSC (2000):
Primary 51A40, 94B27; Secondary 05E20, 05B25, 17A35, 51A35, 51A50
Published electronically:
October 9, 2003
MathSciNet review:
1984461
Fulltext PDF Free Access
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Abstract: There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and linear Kerdock and Preparata codes on the other. These interrelationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of nonisotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of ``boring'' affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes ``boring'' in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.
 [Al1]
A.
A. Albert, Quasigroups. I, Trans. Amer. Math. Soc. 54 (1943), 507–519. MR 0009962
(5,229c), http://dx.doi.org/10.1090/S00029947194300099627
 [Al2]
A.
A. Albert, Finite division algebras and finite planes, Proc.
Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence,
R.I., 1960, pp. 53–70. MR 0116036
(22 #6831)
 [Al3]
A.
A. Albert, Isotopy for generalized twisted fields, An. Acad.
Brasil. Ci. 33 (1961), 265–275. MR 0139639
(25 #3070)
 [Ba]
Reinhold
Baer, Polarities in finite projective
planes, Bull. Amer. Math. Soc. 52 (1946), 77–93. MR 0015219
(7,387d), http://dx.doi.org/10.1090/S000299041946085067
 [BB]
S. Ball and M. R. Brown, The six semifield planes associated with a semifield flock (preprint).
 [Br]
Edgar
H. Brown Jr., Generalizations of the Kervaire invariant, Ann.
of Math. (2) 95 (1972), 368–383. MR 0293642
(45 #2719)
 [CCKS]
A.
R. Calderbank, P.
J. Cameron, W.
M. Kantor, and J.
J. Seidel, 𝑍₄Kerdock codes, orthogonal spreads, and
extremal Euclidean linesets, Proc. London Math. Soc. (3)
75 (1997), no. 2, 436–480. MR 1455862
(98i:94039), http://dx.doi.org/10.1112/S0024611597000403
 [Ch]
Chris
Charnes, A nonsymmetric translation plane of order 17²,
J. Geom. 37 (1990), no. 12, 77–83. MR 1041980
(91g:51009), http://dx.doi.org/10.1007/BF01230360
 [ChD]
C.
Charnes and U.
Dempwolff, The translation planes of order 49 and
their automorphism groups, Math. Comp.
67 (1998), no. 223, 1207–1224. MR 1468940
(98j:51007), http://dx.doi.org/10.1090/S0025571898009612
 [CW]
M.
Cordero and G.
P. Wene, A survey of finite semifields, Discrete Math.
208/209 (1999), 125–137. Combinatorics (Assisi,
1996). MR
1725526 (2001f:12015), http://dx.doi.org/10.1016/S0012365X(99)000680
 [De]
P.
Dembowski, Finite geometries, Ergebnisse der Mathematik und
ihrer Grenzgebiete, Band 44, SpringerVerlag, Berlin, 1968. MR 0233275
(38 #1597)
 [Di]
J. F. Dillon, Elementary Hadamard difference sets. Ph.D. thesis, U. of Maryland 1974.
 [Dy]
R.
H. Dye, Partitions and their stabilizers for line complexes and
quadrics, Ann. Mat. Pura Appl. (4) 114 (1977),
173–194. MR 0493729
(58 #12698)
 [Ga]
M.
J. Ganley, Polarities in translation planes, Geometriae
Dedicata 1 (1972), no. 1, 103–116. MR 0307036
(46 #6157)
 [HK]
Marshall
Hall Jr. and D.
E. Knuth, Combinatorial analysis and computers, Amer. Math.
Monthly 72 (1965), no. 2, 21–28. MR 0172812
(30 #3030)
 [HKCSS]
A.
Roger Hammons Jr., P.
Vijay Kumar, A.
R. Calderbank, N.
J. A. Sloane, and Patrick
Solé, The 𝑍₄linearity of Kerdock, Preparata,
Goethals, and related codes, IEEE Trans. Inform. Theory
40 (1994), no. 2, 301–319. MR 1294046
(95k:94030), http://dx.doi.org/10.1109/18.312154
 [Ka1]
W.
M. Kantor, Spreads, translation planes and Kerdock sets. II,
SIAM J. Algebraic Discrete Methods 3 (1982), no. 3,
308–318. MR
666856 (83m:51013b), http://dx.doi.org/10.1137/0603032
 [Ka2]
William
M. Kantor, An exponential number of generalized Kerdock codes,
Inform. and Control 53 (1982), no. 12, 74–80.
MR 715523
(85i:94022), http://dx.doi.org/10.1016/S00199958(82)911391
 [Ka3]
William
M. Kantor, Codes, quadratic forms and finite geometries,
Different aspects of coding theory (San Francisco, CA, 1995) Proc.
Sympos. Appl. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1995,
pp. 153–177. MR 1368640
(96m:94010)
 [Ka4]
William
M. Kantor, Projective planes of order 𝑞 whose collineation
groups have order 𝑞², J. Algebraic Combin.
3 (1994), no. 4, 405–425. MR 1293823
(96a:51003), http://dx.doi.org/10.1023/A:1022464027664
 [Ka5]
William
M. Kantor, Orthogonal spreads and translation planes, Progress
in algebraic combinatorics (Fukuoka, 1993) Adv. Stud. Pure Math.,
vol. 24, Math. Soc. Japan, Tokyo, 1996, pp. 227–242. MR 1414469
(97i:51017)
 [Ka6]
W. M. Kantor, Commutative semifields and symplectic spreads (to appear in J. Algebra).
 [Ke]
A.
M. Kerdock, A class of lowrate nonlinear binary codes,
Information and Control 20 (1972), 182–187; ibid. 21
(1972), 395. MR
0345707 (49 #10438)
 [Kn1]
Donald
E. Knuth, Finite semifields and projective planes, J. Algebra
2 (1965), 182–217. MR 0175942
(31 #218)
 [Kn2]
Donald
E. Knuth, A class of projective planes,
Trans. Amer. Math. Soc. 115 (1965), 541–549. MR 0202041
(34 #1916), http://dx.doi.org/10.1090/S0002994719650202041X
 [KW]
William
M. Kantor and Michael
E. Williams, New flagtransitive affine planes of even order,
J. Combin. Theory Ser. A 74 (1996), no. 1,
1–13. MR
1383501 (97e:51012), http://dx.doi.org/10.1006/jcta.1996.0033
 [Ma]
A. Maschietti, Symplectic translation planes and line ovals. Adv. Geom. 3 (2003) 123143.
 [MR]
Rudolf
Mathon and Gordon
F. Royle, The translation planes of order 49, Des. Codes
Cryptogr. 5 (1995), no. 1, 57–72. MR 1313377
(95j:51016), http://dx.doi.org/10.1007/BF01388504
 [Pr]
Franco
P. Preparata, A class of optimum nonlinear doubleerrorcorrecting
codes, Information and Control 13 (1968),
378–400. MR 0242563
(39 #3894)
 [Ta]
Donald
E. Taylor, The geometry of the classical groups, Sigma Series
in Pure Mathematics, vol. 9, Heldermann Verlag, Berlin, 1992. MR 1189139
(94d:20028)
 [Wa]
R.
J. Walker, Determination of division algebras with 32
elements, Proc. Sympos. Appl. Math., Vol. XV, Amer. Math. Soc.,
Providence, R.I., 1963, pp. 83–85. MR 0157991
(28 #1219)
 [Wi]
M. E. Williams, Linear Kerdock codes, orthogonal geometries, and nonassociative division algebras. Ph.D. thesis, U. of Oregon 1995.
 [Al1]
 A. A. Albert, Quasigroups I. Trans. AMS 54 (1943) 507519. MR 5:229c
 [Al2]
 A. A. Albert, Finite division algebras and finite planes, pp. 5370 in: AMS Proc. Symp. Appl. Math. 10, 1960. MR 22:6831
 [Al3]
 A. A. Albert, Isotopy for generalized twisted fields. An. Acad. Brasil. Ci. 33 (1961) 265275. MR 25:3070
 [Ba]
 R. Baer, Polarities in finite projective planes. Bull. AMS 52 (1946) 7793. MR 7:387d
 [BB]
 S. Ball and M. R. Brown, The six semifield planes associated with a semifield flock (preprint).
 [Br]
 E. H. Brown, Generalizations of Kervaire's invariant. Annals of Math. 95 (1972) 368383. MR 45:2719
 [CCKS]
 A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, Kerdock codes, orthogonal spreads, and extremal Euclidean linesets. Proc. LMS 75 (1997) 436480. MR 98i:94039
 [Ch]
 C. Charnes, A nonsymmetric translation plane of order . J. Geometry 37 (1990) 7783. MR 91g:51009
 [ChD]
 C. Charnes and U. Dempwolff, The translation planes of order 49 and their automorphism groups. Math. Comp. 67 (1998) 12071224. MR 98j:51007
 [CW]
 M. Cordero and G. P. Wene, A survey of finite semifields. Discrete Math. 208/209 (1999) 125137. MR 2001f:12015
 [De]
 P. Dembowski, Finite geometries. Springer, BerlinHeidelbergNY 1968. MR 38:1597
 [Di]
 J. F. Dillon, Elementary Hadamard difference sets. Ph.D. thesis, U. of Maryland 1974.
 [Dy]
 R. H. Dye, Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl. 114 (1977) 173194. MR 58:12698
 [Ga]
 M. J. Ganley, Polarities in translation planes. Geom. Dedicata 1 (1972) 103116. MR 46:6157
 [HK]
 M. Hall, Jr. and D. E. Knuth, Combinatorial analysis and computers. Amer. Math. Monthly 72 (1965) 2128. MR 30:3030
 [HKCSS]
 A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40 (1994) 301319. MR 95k:94030
 [Ka1]
 W. M. Kantor, Spreads, translation planes and Kerdock sets. I, II. SIAM J. Alg. Discr. Meth. 3 (1982) 151165 and 308318. MR 83m:51013b
 [Ka2]
 W. M. Kantor, An exponential number of generalized Kerdock codes. Inform. Control 53 (1982) 7480. MR 85i:94022
 [Ka3]
 W. M. Kantor, Codes, quadratic forms and finite geometries, pp. 153177 in: Different aspects of coding theory (Ed. A. R. Calderbank), Proc. AMS Symp. Applied Math. 50, 1995. MR 96m:94010
 [Ka4]
 W. M. Kantor, Projective planes of order whose collineation groups have order . J. Alg. Combin. 3 (1994) 405425. MR 96a:51003
 [Ka5]
 W. M. Kantor, Orthogonal spreads and translation planes, pp. 227242 in: Progress in Algebraic Combinatorics (Eds. E. Bannai and A. Munemasa), Advanced Studies in Pure Mathematics 24, Mathematical Society of Japan 1996. MR 97i:51017
 [Ka6]
 W. M. Kantor, Commutative semifields and symplectic spreads (to appear in J. Algebra).
 [Ke]
 A. M. Kerdock, A class of lowrate nonlinear binary codes. Inform. Control 20 (1972) 182187. MR 49:10438
 [Kn1]
 D. E. Knuth, Finite semifields and projective planes. J. Algebra 2 (1965) 182217. MR 31:218
 [Kn2]
 D. E. Knuth, A class of projective planes. Trans. AMS 115 (1965) 541549. MR 34:1916
 [KW]
 W. M. Kantor and M. E. Williams, New flagtransitive affine planes of even order. J. Comb. Theory (A) 74 (1996) 113. MR 97e:51012
 [Ma]
 A. Maschietti, Symplectic translation planes and line ovals. Adv. Geom. 3 (2003) 123143.
 [MR]
 R. Mathon and G. Royle, The translation planes of order 49. Des. Codes Crypt. 5 (1995) 5772. MR 95j:51016
 [Pr]
 F. P. Preparata, A class of optimum nonlinear doubleerror correcting codes. Inform. Control 13 (1968) 378400. MR 39:3894
 [Ta]
 D. E. Taylor, The geometry of the classical groups. Heldermann, Berlin 1992. MR 94d:20028
 [Wa]
 R. J. Walker, Determination of division algebras with elements, pp. 8385 in: Proc. AMS Symp. Applied Math. 15, 1962. MR 28:1219
 [Wi]
 M. E. Williams, Linear Kerdock codes, orthogonal geometries, and nonassociative division algebras. Ph.D. thesis, U. of Oregon 1995.
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Additional Information
William M. Kantor
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email:
kantor@math.uoregon.edu
Michael E. Williams
Affiliation:
Raytheon, Dallas, Texas 75042
Email:
Michael_E1_Williams@raytheon.com
DOI:
http://dx.doi.org/10.1090/S0002994703034019
PII:
S 00029947(03)034019
Keywords:
Projective planes,
semifields,
spreads,
finite orthogonal geometries,
$\mathbb{Z}_4$codes
Received by editor(s):
May 29, 2002
Published electronically:
October 9, 2003
Additional Notes:
This research was supported in part by the National Science Foundation.
Dedicated:
In memory of Jaap Seidel
Article copyright:
© Copyright 2003 American Mathematical Society
