MinT - Best Known (81, 127, s)-Nets in Base 16

Best Known (81, 127, s)-Nets in Base 16

(81, 127, 585)-Net over F₁₆ — Constructive and digital

Digital (81, 127, 585)-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 (52, 98, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 49, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(81, 127, 590)-Net in Base 16 — Constructive

(81, 127, 590)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (12, 35, 76)-net in base 16, using
  - base change [i] based on digital (5, 28, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]
2. digital (46, 92, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 46, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(81, 127, 2963)-Net over F₁₆ — Digital

Digital (81, 127, 2963)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 127, 2800267)-Net in Base 16 — Upper bound on s

There is no (81, 127, 2800268)-net in base 16, because

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]