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Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4. (English) Zbl 1355.94094

Kyureghyan, Gohar (ed.) et al., Topics in finite fields. 11th international conference on finite fields and their applications (Fq11), Magdeburg, Germany, July 22–26, 2013. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9860-4/pbk; 978-1-4704-2220-2/ebook). Contemporary Mathematics 632, 157-176 (2015).
Summary: It is shown that the maximum size of a binary subspace code of packet length \(v=6\), minimum subspace distance \(d=44\), and constant dimension \(k=3\) is \(M=77\); in Finite Geometry terms, the maximum number of planes in \(\mathrm{PG}(5,2)\) mutually intersecting in at most a point is 77. Optimal binary \((v,M,d;k)=(6,77,4;3)\) subspace codes are classified into 5 isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any \(q\), yielding a new family of \(q\)-ary \((6,q^6+2q^2+2q+1,4;3)\) subspace codes.
For the entire collection see [Zbl 1309.11002].

MSC:

94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
05B25 Combinatorial aspects of finite geometries
51E22 Linear codes and caps in Galois spaces
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