Combinatorial Aspects of Code Loops
نویسنده
چکیده
The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T.Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire contained in [2]. Within this paper, we focus on their combinatorial construction and prove a more general result 2.1 using the language of derived forms.
منابع مشابه
Code Loops in Both Parities
We present equivalent definitions of code loops in any characteristic p 6= 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.
متن کاملJa n 20 07 COMBINATORIAL ASPECTS OF CODE LOOPS
The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T.Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E....
متن کاملCombinatorial polarization, code loops, and codes of high level
We first find the combinatorial degree of any map f : V → F where F is a finite field and V is a finite-dimensional vector space over F . We then simplify and generalize a certain construction due to Chein and Goodaire that was used in characterizing code loops as finite Moufang loops that posses at most two squares. The construction yields binary codes of high divisibility level with prescribe...
متن کاملSome combinatorial aspects of finite Hamiltonian groups
In this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite Hamiltonian group. The coverings with three proper subgroups and the principal series of such a group are also counted. Finally, we give a complete description of the lattice of characteristic subgroups of a finite H...
متن کاملCombinatorial Designs and Code Synchronization
Combinatorial Designs and Code Synchronization – p. 1/2
متن کامل