MinT - Best Known (51, 51+29, s)-Nets in Base 16

Best Known (51, 51+29, s)-Nets in Base 16

(51, 51+29, 581)-Net over F₁₆ — Constructive and digital

Digital (51, 80, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 20, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (31, 60, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(51, 51+29, 2090)-Net over F₁₆ — Digital

Digital (51, 80, 2090)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+29, 2512130)-Net in Base 16 — Upper bound on s

There is no (51, 80, 2512131)-net in base 16, because

1 times m-reduction [i] would yield (51, 79, 2512131)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133499 372611 946218 306968 697397 846939 566217 340430 638164 819840 003148 881116 579087 079582 549701 164736 > 16⁷⁹ [i]