Equidistant Linear Codes in Projective Spaces
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Linear codes in the projective space ℙ_q(n), the set of all subspaces of the vector space 𝔽_q^n, were first considered by Braun, Etzion and Vardy. The Grassmannian 𝔾_q(n,k) is the collection of all subspaces of dimension k in ℙ_q(n). We study equidistant linear codes in ℙ_q(n) in this paper and establish that the normalized minimum distance of a linear code is maximum if and only if it is equidistant. We prove that the upper bound on the size of such class of linear codes is 2^n when q=2 as conjectured by Braun et al. Moreover, the codes attaining this bound are shown to have structures akin to combinatorial objects, viz. Fano plane and sunflower. We also prove the existence of equidistant linear codes in ℙ_q(n) for any prime power q using Steiner triple system. Thus we establish that the problem of finding equidistant linear codes of maximum size in ℙ_q(n) with constant distance 2d is equivalent to the problem of finding the largest d-intersecting family of subspaces in 𝔾_q(n, 2d) for all 1 ≤ d ≤⌊n/2⌋. Our discovery proves that there exist equidistant linear codes of size more than 2^n for every prime power q > 2.
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