MinT - Best Known (68−47, 68, s)-Nets in Base 256

Best Known (68−47, 68, s)-Nets in Base 256

(68−47, 68, 278)-Net over F₂₅₆ — Constructive and digital

Digital (21, 68, 278)-net over F₂₅₆, using

net from sequence [i] based on digital (21, 277)-sequence over F₂₅₆, using
- Niederreiter sequence [i]

(68−47, 68, 513)-Net over F₂₅₆ — Digital

Digital (21, 68, 513)-net over F₂₅₆, using

t-expansion [i] based on digital (8, 68, 513)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(68−47, 68, 383011)-Net in Base 256 — Upper bound on s

There is no (21, 68, 383012)-net in base 256, because

1 times m-reduction [i] would yield (21, 67, 383012)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 224959 171263 600955 781239 131105 665142 304424 566319 988265 588427 907775 165786 829383 751772 833762 795764 909144 651171 383169 431849 365066 701868 594794 555799 229699 781113 714856 > 256⁶⁷ [i]