Linearly embeddable designs

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A residual design DB with respect to a block B of a given design D is defined to be linearly embeddable over GF(p) if the p-ranks of the incidence matrices of DB and D differ by one. A sufficient condition for a residual design to be linearly embeddable is proved in terms of the minimum distance of the linear code spanned by the incidence matrix, and this condition is used to show that the residual designs of several known infinite classes of designs are linearly embeddable. A necessary condition for linear embeddability is proved for affine resolvable designs and their residual designs. As an application, it is shown that a residual design of the classical affine design of the planes in AG (3,22) admits two nonisomorphic embeddings over GF(2) that give rise to the only known counterexamples to Hamada’s conjecture over a field of non-prime order.

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© Springer Science+Business Media New York 2016. Publisher’s version of record: https://doi.org/10.1007/s10623-016-0304-6

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Designs, Codes and Cryptography