TY - JOUR

T1 - Sparse hypergraphs with applications to coding theory

AU - Shangguan, Chong

AU - Tamo, Itzhak

N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics

PY - 2020

Y1 - 2020

N2 - For fixed integers r \geq 3, e \geq 3, v \geq r + 1, an r-uniform hypergraph is called Gr(v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erd\H os, and Sos showed that the maximum number of edges of such a hypergraph on n vertices, denoted as fr(n, v, e), satisfies \Omega (n er e - -1 v ) = fr(n, v, e) = O(n\lceil er e - -1 v \rceil ). For sufficiently large n and e - 1 | er - v, the lower bound matches the upper bound up to a constant factor, which depends only on r, v, e; whereas for e - 1 \nmid er - v, in general it is a notoriously hard problem to determine the correct exponent of n. Among other results, we improve the above lower bound by showing that fr(n, v, e) = \Omega (n er e - -1 v (log n) e -1 1 ) for any r, e, v satisfying gcd(e - 1, er - v) = 1. The hypergraph we constructed is in fact Gr(ir - \lceil (i - 1)(er - v) \rceil, i)-free for every 2 \leq i \leq e, and it has several interesting e - 1 applications in coding theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and Rodl.

AB - For fixed integers r \geq 3, e \geq 3, v \geq r + 1, an r-uniform hypergraph is called Gr(v, e)-free if the union of any e distinct edges contains at least v + 1 vertices. Brown, Erd\H os, and Sos showed that the maximum number of edges of such a hypergraph on n vertices, denoted as fr(n, v, e), satisfies \Omega (n er e - -1 v ) = fr(n, v, e) = O(n\lceil er e - -1 v \rceil ). For sufficiently large n and e - 1 | er - v, the lower bound matches the upper bound up to a constant factor, which depends only on r, v, e; whereas for e - 1 \nmid er - v, in general it is a notoriously hard problem to determine the correct exponent of n. Among other results, we improve the above lower bound by showing that fr(n, v, e) = \Omega (n er e - -1 v (log n) e -1 1 ) for any r, e, v satisfying gcd(e - 1, er - v) = 1. The hypergraph we constructed is in fact Gr(ir - \lceil (i - 1)(er - v) \rceil, i)-free for every 2 \leq i \leq e, and it has several interesting e - 1 applications in coding theory. The proof of the new lower bound is based on a novel application of the lower bound on the hypergraph independence number due to Duke, Lefmann, and Rodl.

KW - Coding theory

KW - Hypergraph independence number

KW - Sparse hypergraphs

UR - http://www.scopus.com/inward/record.url?scp=85092265992&partnerID=8YFLogxK

U2 - 10.1137/19M1248108

DO - 10.1137/19M1248108

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AN - SCOPUS:85092265992

SN - 0895-4801

VL - 34

SP - 1493

EP - 1504

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

ER -