A modular equality for Cameron–Liebler line classes

Cameron‐Liebler line classes in PG(3, 5)

“…equations, however, it appears to be hard to determine its rank (even calculating the precise values of | | M , | | N can be very tedious, see [14,Section 3]), as its coefficients depend on the structure of all patterns.…”

“…Thus, if q PG(3, ) has a Cameron-Liebler line class with parameter x, then Equation (1) has a solution for n in the set q {0, 1, …, }. (see [14,Section 3]). However, it gives only a necessary but not sufficient criterion for the existence of Cameron-Liebler line classes: for example, in PG(3, 5), for ∈ x {4, 5, 6, 8, 9, 10, 12, 13}, Equation (1) has a solution in n, however, line classes with parameter ∈…”

“…• with x = 10 constructed in [14]; • with x = 12 constructed in [25] (later generalised in [7,10], to an infinite family with x = q − 1 2 2 for ≡ q 5 or 9 (mod 12));…”

Cameron–Liebler line classes with parameter x=q2−12

“…Later on it was found that these line classes have many connections to other geometric and combinatorial objects, such as blocking sets of PG(2, q), projective two-intersection sets in PG (5, q), two-weight linear codes, and strongly regular graphs. In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics; see, for example, [7,20,21,26,11,10]. In [6], the authors gave several equivalent conditions for a set of lines of PG (3, q) to be a Cameron-Liebler line class; Penttila [23] gave a few more of such characterizations.…”

“…It seems reasonable to believe that for any fixed 0 < < 1 and constant c > 0 there are no CameronLiebler line classes with 2 < x < cq 2− for sufficiently large q. Very recently, Gavrilyuk and Metsch [10] proved a modular equality which eliminates almost half of the possible values x for a Cameron-Liebler line class with parameter x. We refer to [21] for a comprehensive survey of the known nonexistence results.…”