Projective systems whose supports consist of the union of three linear subspaces (Q2770334)

Let \(\text{GF}(q)\) be the finite field with \(q\) elements. Let \(C\subset \text{GF}(q)^n\) be a non-degenerate \(q\)-ary \([n,k,d]\)-code with generator matrix \(G=(a_1,\dots,a_n)\), each \(a_i\) being a column vector of length \(k\). Then \(G\) induces a positive \(0\)-cycle on \(\text{PG}(k-1,q)\), whose support spans the whole space. On the other hand a positive \(0\)-cycle of length \(n\) in \(\text{PG}(k-1,q)\) whose support spans \(\text{PG}(k-1,q)\) gives rise to a non-degenerate \([n,k]_q\)-code. Let \(S\subseteq \text{PG}(k-1,q)\) span the whole space. Then, the \(0\)-cycle \(\sum_{P\in S}P\) induces a code \(C\). In [Bull. Korean Math. Soc. 37, 493-505 (2000; Zbl 0963.94036)], \textit{M. Homma, S. J. Kim} and \textit{M. Y. Yoo} computed the weight enumerator \(W_C(z)\) of \(C\) when \(S\) is a union of subspaces, \(L_i\), \(i=1,\dots r\), of \(\text{PG}(k-1,q)\) in general position. Assuming \(r=2\), Homma, Kim and Yoo proved the converse: if \(W_C(z)=W_{C'}(z)\), then \(C'\) is equivalent to \(C\). In the paper under review the authors prove the converse in the case \(r=3\).