MinT - Best Known (78, 78+47, s)-Nets in Base 16

Best Known (78, 78+47, s)-Nets in Base 16

(78, 78+47, 581)-Net over F₁₆ — Constructive and digital

Digital (78, 125, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 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 (49, 96, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 48, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(78, 78+47, 2268)-Net over F₁₆ — Digital

Digital (78, 125, 2268)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+47, 1950471)-Net in Base 16 — Upper bound on s

There is no (78, 125, 1950472)-net in base 16, because

1 times m-reduction [i] would yield (78, 124, 1950472)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 204588 999926 929031 598299 705115 048862 137701 463349 940983 788079 235943 423427 107991 243875 924709 420488 634607 958104 567434 207809 868573 066471 368612 648775 509616 > 16¹²⁴ [i]