Construction of quantum caps in projective space \(\mathrm{PG}(r,4)\) and quantum codes of distance 4. (English) Zbl 1333.81102
Summary: Constructions of quantum caps in projective space \(\mathrm{PG}(r,4)\) by recursive methods and computer search are discussed. For each even \(n\) satisfying \(n\geq 282\) and each odd \(z\) satisfying \(z\geq 275\), a quantum \(n\)-cap and a quantum \(z\)-cap in \(\mathrm{PG}(k-1,4)\) with suitable \(k\) are constructed, and \([[n,n-2k,4]]\) and \([[z,z-2k,4]]\) quantum codes are derived from the constructed quantum \(n\)-cap and \(z\)-cap, respectively. For \(n\geq 282\) and \(n\neq 286\), 756 and 5040, or \(z\geq 275\), the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in \(\mathrm{PG}(r,4)\) for \(r\geq 11\). These results concerning large caps provide improved lower bounds on the maximal sizes of caps in \(\mathrm{PG}(r,4)\) for \(r\geq 11\).
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