MinT - Best Known (78, 78+45, s)-Nets in Base 16

Best Known (78, 78+45, s)-Nets in Base 16

(78, 78+45, 583)-Net over F₁₆ — Constructive and digital

Digital (78, 123, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (77, 122, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 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, 94, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 47, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(78, 78+45, 2694)-Net over F₁₆ — Digital

Digital (78, 123, 2694)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+45, 2871789)-Net in Base 16 — Upper bound on s

There is no (78, 123, 2871790)-net in base 16, because

1 times m-reduction [i] would yield (78, 122, 2871790)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 799 171256 818505 455665 646771 088520 696428 578332 168236 588479 433480 244512 531277 243681 971680 768847 805211 669001 740664 448905 355852 529606 428756 266496 169576 > 16¹²² [i]