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I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most $r$ rows. But if $r$ is less than $n/k$, where $k$ is smallest such that $kw$ is a (positive) multiple of $n,$ then (using that $A$ is cyclic) there is a set of $k$ rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.

I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most $r$ rows. But if $r$ is less than $n/k$, where $k$ is smallest such that $kw$ is greater than and a multiple of $n,$ then (using that $A$ is cyclic) there is a set of $k$ rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.

I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most r rows. But if r is less than n/k, where k is smallest such that kw is a multiple of n, then (using that $A$ is cyclic) there is a set of k rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.

I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most $r$ rows. But if $r$ is less than $n/k$, where $k$ is smallest such that $kw$ is a (positive) multiple of $n,$ then (using that $A$ is cyclic) there is a set of $k$ rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.

I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from A in at most r rows. But if r is less than n/k, where k is smallest such that kw is a multiple of n, then (using that $A$ is cyclic) there is a set of k rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.

I've decided to expand upon the observations on the comments.

I haven't wrapped my head yet around the idea that $A-BC$ is a 0-1 matrix. Thus I assume you or someone else has a proof for that part. Then this matrix differs from $A$ in at most r rows. But if r is less than n/k, where k is smallest such that kw is a multiple of n, then (using that $A$ is cyclic) there is a set of k rows of $A-BC$ which add up to a nontrivial multiple of the row of all ones, and thus $A$ and $A-BC$ have determinants which are not one. This handles some of the cases and reveals some of the number theory going on here.

Gerhard "Number Theory To The Rescue?" Paseman, 2019.03.05.