MinT - Best Known (58, 58+38, s)-Nets in Base 16

Best Known (58, 58+38, s)-Nets in Base 16

(58, 58+38, 538)-Net over F₁₆ — Constructive and digital

Digital (58, 96, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (38, 76, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 38, 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]

(58, 58+38, 1319)-Net over F₁₆ — Digital

Digital (58, 96, 1319)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+38, 641345)-Net in Base 16 — Upper bound on s

There is no (58, 96, 641346)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 402775 033767 390945 871385 423002 410408 508586 540392 918839 785451 913715 272097 320356 336556 827096 345306 670530 363491 242336 > 16⁹⁶ [i]