MinT - Best Known (50, 50+51, s)-Nets in Base 16

Best Known (50, 50+51, s)-Nets in Base 16

(50, 50+51, 257)-Net over F₁₆ — Constructive and digital

Digital (50, 101, 257)-net over F₁₆, using

base reduction for projective spaces (embedding PG(50,256) in PG(100,16)) for nets [i] based on digital (0, 51, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(50, 50+51, 345)-Net over F₁₆ — Digital

Digital (50, 101, 345)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+51, 44450)-Net in Base 16 — Upper bound on s

There is no (50, 101, 44451)-net in base 16, because

1 times m-reduction [i] would yield (50, 100, 44451)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 582328 932168 874354 553136 557739 072771 538041 690929 774678 751395 005223 748612 409354 437939 227652 797003 966203 763576 728474 030376 > 16¹⁰⁰ [i]