MinT - Best Known (56, 56+57, s)-Nets in Base 16

Best Known (56, 56+57, s)-Nets in Base 16

(56, 56+57, 257)-Net over F₁₆ — Constructive and digital

Digital (56, 113, 257)-net over F₁₆, using

base reduction for projective spaces (embedding PG(56,256) in PG(112,16)) for nets [i] based on digital (0, 57, 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]

(56, 56+57, 380)-Net over F₁₆ — Digital

Digital (56, 113, 380)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+57, 49346)-Net in Base 16 — Upper bound on s

There is no (56, 113, 49347)-net in base 16, because

1 times m-reduction [i] would yield (56, 112, 49347)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 726 944737 598273 077600 402769 207240 034127 204653 831134 548306 711627 461801 653045 243652 347109 155652 047931 987254 486635 852049 885938 212182 640016 > 16¹¹² [i]