Code loops in both parities
Aleš Drápal
and Petr Vojtěchovský
DOI: 10.1007/s10801-009-0203-2
Abstract
We present equivalent definitions of code loops in any characteristic p\neq 0. The most natural definition is via combinatorial polarization, but we also show how to realize code loops by linear codes and as a class of symplectic conjugacy closed loops. For p odd, it is possible to define code loops via characteristic trilinear forms. Related concepts are discussed.
Pages: 585–611
Keywords: keywords code loop; even code loop; odd code loop; symplectic loop; small Frattini loop; Moufang loop; conjugacy closed loop; combinatorial polarization; symmetric associator; characteristic form; doubly even code; self-orthogonal code; Kronecker product
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20. Springer, Berlin (1971)
3. Chein, O., Goodaire, E.G.: Moufang loops with a unique nonidentity commutator (associator, square). J. Algebra 130, 369-384 (1990)
4. Chein, O., Robinson, D.A.: An “extra” law for characterizing Moufang loops. Proc. Amer. Math. Soc. 33, 29-32 (1972)
5. Conway, J.H.: A simple construction for the Fischer-Griess Monster group. Invent. Math. 79, 513-540 (1985)
6. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, third edition. Grundlehren der mathematischen Wissenschaften, vol.
290. Springer, Berlin (1999)
7. Csörg\Acute\Acute o, P., Drápal, A.: Left conjugacy closed loops of nilpotency class two. Results Math. 47(3-4), 242-265 (2005)
8. Drápal, A.: On extraspecial left conjugacy closed loops. J. Algebra 302, 771-792 (2006)
9. Drápal, A., Vojt\check echovský, P.: Symmetric multilinear forms and polarization of polynomials. Linear Algebra Appl. 431, 998-1012 (2009)
10. Eilenberg, S., MacLane, S.: Algebraic cohomology groups and loops. Duke Math. J. 14, 435-463 (1947)
11. Ferrero, M., Micali, A.: Sur les n-applications. Coll. Formes Quadratiques 2 (Montpellier, 1977), Bull. Soc. Math. France Mém. No. 59, 33-53 (1979)
12. Glauberman, G.: On loops of odd order II. J. Algebra 8, 393-414 (1968)
13. Glauberman, G., Wright, C.R.B.: Nilpotence of finite Moufang 2-loops. J. Algebra 8, 415-417 (1968)
14. Griess, R.L. Jr.: A construction of F1 as automorphisms of a 196, 883-dimensional algebra. Proc. Nat. Acad. Sci. 78, 689-691 (1981)
15. Griess, R.L. Jr.: The friendly giant. Invent. Math. 69, 1-102 (1982)
16. Griess, R.L. Jr.: Code loops. J. Algebra 100, 224-234 (1986)
17. Hsu, T.: Moufang loops of class 2 and cubic forms. Math. Proc. Cambridge Philos. Soc. 128(2), 197- 222 (2000)
18. Hsu, T.: Explicit constructions of code loops as centrally twisted products. Math. Proc. Cambridge Philos. Soc. 128(2), 223-232 (2000)
19. Johnson, K.W., Leedham-Green, C.R.: Loop cohomology. Czechoslovak Math. J. 40(115(2)), 182- 194 (1990)
20. Kinyon, M.K., Kunen, K.: Power-associative, conjugacy closed loops. J. Algebra 304(2), 679-711 (2006)
21. Kinyon, M.K., Jones, O.: Loops and semidirect products. Comm. Algebra 28(9), 4137-4164 (2000)
22. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Dover, New York (1992)
23. Nagy, G.P.: Direct construction of code loops. Discrete Math. 308(23), 5349-5357 (2008)
24. Nagy, G.P., Vojt\check echovský, P.: The Moufang loops of order 64 and
81. J. Symbolic Comput. 42(9), 871-883 (2007)
25. Nagy, P., Strambach, K.: Schreier loops. Czechoslovak Math. J. 58(133(3)), 759-786 (2008)
26. Pflugfelder, H.O.: Quasigroups and Loops: Introduction. Sigma Series in Pure Mathematics, vol.
7. Heldermann, Berlin (1990)
27. Prószyński, A.: Forms and mappings. I. Generalities. Fund. Math. 122(3), 219-235 (1984)
28. Prószyński, A.: Forms and mappings. IV. Degree
4. Bull. Polish Acad. Sci. Math. 37(1-6), 269-278 (1989)
29. Richardson, T.M.: Local subgroups of the Monster and odd code loops. Trans. Amer. Math. Soc. 347(5), 1453-1531 (1995)
30. Vojt\check echovský, P.: Combinatorial polarization, code loops, and codes of high level. Int. J. Math. Math.
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