2021/07/22 by Pranab Basu, Basu, Pranab
Computer Science · Mathematics · #05B07 (Secondary) #05D05 (Primary) 05B05 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.DM #cs.IT #math.CO #math.IT #msc:05B05 #msc:05B07 #msc:05D05
paper · pdf · doi:10.48550/arxiv.2107.10820
13 pages, 1 figure, submitted to Designs, Codes and Cryptography
arxiv created 2021/07/22 · arxiv updated 2021/07/23
Linear codes in the projective space ℙq(n), the set of all subspaces of the vector space \mathbbFqn, were first considered by Braun, Etzion and Vardy. The Grassmannian \mathbbGq(n,k) is the collection of all subspaces of dimension k in ℙq(n). We study equidistant linear codes in ℙq(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 2n 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 ℙq(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 ℙq(n) with constant distance 2d is equivalent to the problem of finding the largest d-intersecting family of subspaces in \mathbbGq(n, 2d) for all 1 ≤ d ≤ \lfloor (n)/(2)\rfloor. Our discovery proves that there exist equidistant linear codes of size more than 2n for every prime power q > 2.