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  "body": "\n## Definition\n\nLet $\\mathcal{P}$ be a [[projective space|ZT9A6JK8QF]] and let $k \\ge 0$ be an integer.\nA *$k$-spread* of $\\mathcal{P}$ is a set $\\mathcal{S}$ of $k$-dimensional [[subspaces|2RD6T5KG59]] such that every point lies in exactly one member of $\\mathcal{S}$.\n\n## Remarks\n\nA $1$-spread is a *line spread*.\nIn the finite space $\\mathrm{PG}(n, q)$ over the [[finite field|0ZXZ82FTX2]] $\\F_q$, a $k$-spread exists if and only if $k + 1$ divides $n + 1$.\n\nExample: let $V = \\F_{q^2}^2$, which is a vector space of dimension $4$ over $\\F_q$, so its $1$-dimensional $\\F_q$-subspaces are the points of $\\mathrm{PG}(3, q)$.\nThe $1$-dimensional $\\F_{q^2}$-subspaces of $V$ are $2$-dimensional over $\\F_q$, so they are lines of $\\mathrm{PG}(3, q)$.\nEvery nonzero vector of $V$ spans exactly one $1$-dimensional $\\F_{q^2}$-subspace, so these $q^2 + 1$ lines form a line spread, called the regular (or Desarguesian) spread.\n\nNon-example: two lines of $\\mathrm{PG}(3, q)$ that meet in a point $P$ cannot both belong to a spread, since $P$ would lie in two members.\n\nGiven a line spread of $\\mathrm{PG}(3, q)$, the André–Bruck–Bose construction produces a [[translation plane|JZ5J6REAMD]] of order $q^2$.\n",
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