The Weight Distribution of a Family of Lagrangian-Grassmannian Codes

Codes on Linear Sections of Grassmannians

“…Since h(L(2, 4)(F q )) = 0, then h = A 14 X 14 + A 23 X 23 , and since X 14 + X 23 = 0, it follows that h = (A 14 − A 23 )X 14 . By [5], w = (1, 0, 1, 0, 1) ∈ L(2, 4)(F q ) and thus h(w) = (A 14 − A 23 )1 = 0, that is A 14 = A 23 =: A, and consequently h = A(X 14 + X 23 ) = AΠ, as required. Our induction hypothesis is: For all k < n, every h ∈ (∧ k E) * such that L(k, 2k)(F q ) ⊆ Z h, Π αrs : α rs ∈ I(k − 2, 2k) , must be of the form h = A αrs Π αrs , for α rs ∈ I(k − 2, 2k).…”

“…, see [4]. For some low dimension Lagrangian-Grassmannian codes their weight spectra have been completely determined, for example, for the Lagrangian-Grassmannian C L(2,4) code, by [6] and [5], see also [3], and for the Lagrangian-Grassmannian C L (3,6) code in [3].…”