430 research outputs found

    Automorphism groups of Grassmann codes

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    We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.Comment: revised versio

    On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4

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    A subset SS of {0,1,...,2t1}n\{0,1,...,2t-1\}^n is called a tt-fold MDS code if every line in each of nn base directions contains exactly tt elements of SS. The adjacency graph of a tt-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of tt-fold MDS codes of smaller lengths. In the case t=2t=2, the theory has the following application. The union of two disjoint (n,4n1,2)(n,4^{n-1},2) MDS codes in {0,1,2,3}n\{0,1,2,3\}^n is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation as a test for reducibility of nn-quasigroups of order 4. Keywords: MDS codes, n-quasigroups, decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Mat

    On the equivalence of linear sets

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    Let LL be a linear set of pseudoregulus type in a line \ell in Σ=PG(t1,qt)\Sigma^*=\mathrm{PG}(t-1,q^t), t=5t=5 or t>6t>6. We provide examples of qq-order canonical subgeometries Σ1,Σ2Σ\Sigma_1,\, \Sigma_2 \subset \Sigma^* such that there is a (t3)(t-3)-space ΓΣ(Σ1Σ2)\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell) with the property that for i=1,2i=1,2, LL is the projection of Σi\Sigma_i from center Γ\Gamma and there exists no collineation ϕ\phi of Σ\Sigma^* such that Γϕ=Γ\Gamma^{\phi}=\Gamma and Σ1ϕ=Σ2\Sigma_1^{\phi}=\Sigma_2. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented. The final version is to appear in Designs, Codes and Cryptograph
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