A new class of binary constant weight codes derived by groups of linear fractional mappings
Let A(n, d, w) denote the maximum possible number of code words in binary (n, d, w) constant weight codes. For smaller instances of (n, d, w)s. many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, Omega f) := ("Group G," "Set Omega," "Action f on Omega") simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include infinity homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL(2)(F-q), P-1 (F-q), "linear fractional action of subgroups of PGL(2)(F-q) on P-1(F-q)") as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating, group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.
binary constant weight codes linear fractional mappings lower bounds of the number of code words permutation representations pgl(2) (fq) modular groups