Abstract
Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q k/2 of the values of d < q k-1. In this article we shall prove that, for q = p prime and roughly \({\frac{3}{8}}\)-th’s of the values of d < q k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \({{\mathbb F}_p}\) of dimension k with minimum distance d < q k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.
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Communicated by J. D. Key.
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Ball, S., Fancsali, S.L. Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes. Des. Codes Cryptogr. 53, 119–136 (2009). https://doi.org/10.1007/s10623-009-9298-7
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DOI: https://doi.org/10.1007/s10623-009-9298-7