MinT - Best Known (81, 128, s)-Nets in Base 16

Best Known (81, 128, s)-Nets in Base 16

(81, 128, 583)-Net over F₁₆ — Constructive and digital

Digital (81, 128, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (80, 127, 583)-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 (51, 98, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 49, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(81, 128, 2713)-Net over F₁₆ — Digital

Digital (81, 128, 2713)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 128, 2800267)-Net in Base 16 — Upper bound on s

There is no (81, 128, 2800268)-net in base 16, because

1 times m-reduction [i] would yield (81, 127, 2800268)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 837 992208 099999 966779 752289 013184 514975 692386 850315 685854 240465 560176 785682 400541 487695 415227 398313 034521 823898 656568 523104 335978 329793 430442 854785 349136 > 16¹²⁷ [i]