Abstract
In this article, we show the existence of large sets \({\text {LS}}_2[3](2,k,v)\) for infinitely many values of k and v. The exact condition is \(v \ge 8\) and \(0 \le k \le v\) such that for the remainders \(\bar{v}\) and \(\bar{k}\) of v and k modulo 6 we have \(2 \le \bar{v} < \bar{k} \le 5\). The proof is constructive and consists of two parts. First, we give a computer construction for an \({\text {LS}}_2[3](2,4,8)\), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-\((8, 4, 217)_2\) subspace designs. Together with the already known \({\text {LS}}_2[3](2,3,8)\), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.
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The authors would like to acknowledge the financial support provided by COST—European Cooperation in Science and Technology. The authors are members of the Action IC1104 Random Network Coding and Designs over GF(q).
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This is one of several papers published in Designs, Codes and Cryptography comprising the Special Issue on Network Coding and Designs.
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Kiermaier, M., Laue, R. & Wassermann, A. A new series of large sets of subspace designs over the binary field. Des. Codes Cryptogr. 86, 251–268 (2018). https://doi.org/10.1007/s10623-017-0349-1
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DOI: https://doi.org/10.1007/s10623-017-0349-1