Linear codes over signed graphs

Abstract

We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual code. Then we determine the regularity of the ideals of circuits and cocircuits of a signed graph, and prove an algebraic formula in terms of the multiplicity for the frustration index of an unbalanced signed graph. IntroductionThe generalized Hamming weights (GHWs) of a linear code are parameters of interest in many applications [12,16,20,27,31,37,42,43,45] and they have been nicely related to the graded Betti numbers of the ideal of cocircuits of the matroid of a linear code [19,20], to the nullity function of the dual matroid of a linear code [42], and to the enumerative combinatorics of linear codes [3,18,22,23]. Because of this, their study has attracted considerable attention, but determining them is in general a difficult problem. The notion of generalized Hamming weight was introduced by Helleseth, Kløve and Mykkeltveit in [17] and was first used systematically by Wei in [42]. For convenience we recall this notion. Let K = F q be a finite field and let C be an [m, k]-linear code of length m and dimension k, that is, C is a linear subspace of K m with k = dim K (C). Let 1 ≤ r ≤ k be an integer. Given a linear subspace D of C, the support of D, denoted χ(D), is the set of nonzero positions of D, that is, χ(D) := {i : ∃ (a 1 , . . . , a m ) ∈ D, a i = 0}.The r-th generalized Hamming weight of C, denoted δ r (C), is given by δ r (C) := min{|χ(D)| : D is a subspace of C, dim K (D) = r}.As usual we call the set {δ 1 (C), . . . , δ k (C)} the weight hierarchy of the linear code C. The 1st Hamming weight of C is the minimum distance δ(C) of C, that is, one haswhere ω(x) is the Hamming weight of the vector x, i.e., the number of non-zero entries of x. To determine the minimum distance is essential to find good error-correcting codes [23].The notion of generalized Hamming weights for linear codes was extended to matroids by Britz, Johnsen, Mayhew and Shiromoto [6, p. 332] as we now explain.Let M be a matroid with ground set E, rank function ρ, nullity function η, and let M * be its dual matroid. The r-th generalized Hamming weight of M , denoted d r (M ), is given by d r (M ) := min{|X| : X ⊆ E and η(X) = r} for 1 ≤ r ≤ η(E).2010 Mathematics Subject Classification. Primary 94B05; Secondary 94C15, 05C40, 05C22; 13P25.

Greedy weights for matroids

“…The weight d 2 corresponds to the minimum support of the union of two different cycles (the right part of the graph), while the weight e 2 corresponds to the minimum cardinality of the support of the union of two different cycles, where one of the cycles is a cycle of minimal length (the left part of the graph, since the only cycle of minimal length is the hexagon on the left part). See also [14] for the computation of generalized Hamming weights of (signed) graphs.…”