MinT - Best Known (130−57, 130, s)-Nets in Base 16

Best Known (130−57, 130, s)-Nets in Base 16

(130−57, 130, 530)-Net over F₁₆ — Constructive and digital

Digital (73, 130, 530)-net over F₁₆, using

trace code for nets [i] based on digital (8, 65, 265)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(130−57, 130, 1026)-Net over F₁₆ — Digital

Digital (73, 130, 1026)-net over F₁₆, using

trace code for nets [i] based on digital (8, 65, 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]

(130−57, 130, 265730)-Net in Base 16 — Upper bound on s

There is no (73, 130, 265731)-net in base 16, because

1 times m-reduction [i] would yield (73, 129, 265731)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 214535 819943 478789 173776 486495 895857 170488 745470 741255 083861 615208 623471 419701 303872 000549 240329 390606 492285 041159 093922 861674 204763 631195 075247 197102 317696 > 16¹²⁹ [i]