Frmo the introduction: Let \(({\mathcal X},{\mathcal L})\) be a partial linear space such that each of its points is contained in at least \(n+1\) lines. Fix a prime \(p\). Let \({\mathcal C}_{\mathcal L}\subseteq \mathbb{F}^{\mathcal X}_p\) denote the \(p\)-ary code generated by lines in \({\mathcal L}\) and \({\mathcal C}^\perp_{\mathcal L}\) denote its dual.
In this article, we prove that the minimum weight of \({\mathcal C}^\perp_{\mathcal L}\) is at least \(2n+ 2-{2n\over p}\). (As a special case of our result we see that the minimum weight of the \(p\)-ary code dual to the code generated by lines in a projective plane of order \(n\) is at least \(2n+ 2-{2n\over p}\). This bound sharply improves upon the earlier known bounds as given in [E. F. Assumus, jun. and J. D. Key, “Designs and their codes”, Cambridge Univ. Press, Cambridge, UK (1992; Zbl 0762.05001), Corollary 6.3.1 and Theorem 6.4.2].) We also prove that the induced structure on the support of a word of weight \(2n+ 2-{2n\over p}\) in \({\mathcal C}^\perp_{\mathcal L}\), if it exists, is isomorphic to the join of two Steiner 2-designs with \({n\over p}\) lines through every point and \(p\) points on every line (Theorem 2).