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Projective systems supported on the complement of two linear subspaces. (English) Zbl 0963.94036

Let \(GF(q)\) be the finite field of order \(q\) and let \(PG(m,q)\) be the \(m\)-dimensional projective space over \(GF(q)\). A code \(C\subset GF(q)^n\) is said to be degenerate if there exists a position \(i\), (\(1\leq i\leq n\)) such that \(c_i=0\) for any codeword \(c=(c_1,\dots,c_n)\in C\). Otherwise \(C\) is said to be nondegenerate.
For a nondegenerate code \(C\) one can define a positive \(0\)-cycle \(\sum_{i=1}^n [a_i]\) on \(PG(k-1,q)\), where \([a_i]\) is the point of the projective space whose representative is \(a_i\). For a given nondegenerate \([n,k]_q\)-code \(C\), one can define a projective equivalence class of positive \(0\)-cycles on \(PG(k-1,q)\) whose supports span the whole space and denote by \({\chi}_C\) one of the \(0\)-cycles. Then, the \(0\)-cycle is called a projective system associated to \(C\).
The authors investigate the class of projective systems whose supports are the complement of the union of two linear subspaces in general position. They express the weight enumerators of the codes generated by these projective systems using two simplex codes corresponding to given linear subspaces.
Finally, they prove that these codes are uniquely determined up to equivalence by their weight enumerators.

MSC:

94B05 Linear codes (general theory)
51E20 Combinatorial structures in finite projective spaces
05B25 Combinatorial aspects of finite geometries
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