MinT - Best Known (57, 57+41, s)-Nets in Base 16

Best Known (57, 57+41, s)-Nets in Base 16

(57, 57+41, 530)-Net over F₁₆ — Constructive and digital

Digital (57, 98, 530)-net over F₁₆, using

trace code for nets [i] based on digital (8, 49, 265)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(57, 57+41, 1026)-Net over F₁₆ — Digital

Digital (57, 98, 1026)-net over F₁₆, using

trace code for nets [i] based on digital (8, 49, 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]

(57, 57+41, 382987)-Net in Base 16 — Upper bound on s

There is no (57, 98, 382988)-net in base 16, because

1 times m-reduction [i] would yield (57, 97, 382988)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 630 448453 106430 944029 637485 005342 930077 597809 058473 389058 249489 561928 807099 929312 605284 115834 080905 836425 425717 739776 > 16⁹⁷ [i]