# reference request – A Hadamard product of binary (or ternary) matroids

I would like to know if anyone has studied the following “Hadamard product” of binary (or ternary) matroids. (There is a notion of Hadamard product of matroids studied e.g. here but I think that one is different.)

Let $$M,N$$ be simple binary matroids of rank $$r$$ and $$s$$, respectively, over the same ground set $$E$$ of size $$n$$. For binary representations $$(x_1,dots, x_n)$$ and $$(y_1,dots, y_n)$$ of $$M$$ and $$N$$, respectively, define the Hadamard product of $$M circ N$$ to be the binary matroid represented by $$(x_1 otimes y_1, dots, x_n otimes y_n)$$. One can easily show that this is a well-defined matroid product, using the fact all representations of binary matroids are projectively equivalent (Proposition 6.6.5, Matroid Theory, Oxley).

After a little work, one can derive the linearly independent sets in $$M circ N$$. Suppose WLOG that $$(x_1,dots, x_r)$$ form a basis for $$M$$. For each $$i in {1,dots, r}$$, let

$$text{Supp}(i)={a in {1,dots, n} | x_a(i) neq 0},$$

where $$x_a(i)in mathbb{F}_2$$ is the $$i$$-th coordinate of $$x_a$$ in the basis $$(x_1,dots, x_r)$$. Then $$S subseteq (n)$$ is linearly independent in $$M circ N$$ if and only if for all $$T subseteq (n)$$ of size $$1 leq |{T}| leq n-1$$, there exists $$i in {1,dots, r}$$ such that $$sum_{a in T} x_a(i) y_a neq 0$$. This inequality is equivalent (over $$mathbb{F}_2$$), to saying that the set

$$T cap text{Supp}(i)$$ is not Eulerian in $$N$$, i.e. it cannot be partitioned into circuits in $$N$$.

As a side note, I would also be very interested in any feedback on the following conjecture, which is the $$mathbb{F}_2$$-version of a conjecture I have been thinking about for some time (preprint here).

Conjecture: Let $$M_1,dots, M_m$$ be simple binary matroids of rank $$r_1,dots, r_m$$, respectively over the same ground set $$E$$ of size $$n$$. If $$n leq sum_{j=1}^m (r_j-1)+1$$, then $$M_1 circ dots circ M_m$$ does not form a circuit.

I have proven this conjecture when $$m=2$$; or $$m=3$$ and $$r_3=2$$; or $$m$$ is arbitrary, $$r_1geq 1$$ is arbitrary, and $$r_2=dots=r_m=2$$.