MinT - Best Known (151, 151+83, s)-Nets in Base 4

Best Known (151, 151+83, s)-Nets in Base 4

(151, 151+83, 163)-Net over F₄ — Constructive and digital

Digital (151, 234, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 56, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 178, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 89, 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]

(151, 151+83, 208)-Net in Base 4 — Constructive

(151, 234, 208)-net in base 4, using

2 times m-reduction [i] based on (151, 236, 208)-net in base 4, using
- trace code for nets [i] based on (33, 118, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (33, 120, 104)-net in base 16, using
    - base change [i] based on digital (9, 96, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(151, 151+83, 507)-Net over F₄ — Digital

Digital (151, 234, 507)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(151, 151+83, 14165)-Net in Base 4 — Upper bound on s

There is no (151, 234, 14166)-net in base 4, because

1 times m-reduction [i] would yield (151, 233, 14166)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 945018 804458 602557 865293 287828 927695 069061 939482 420870 286865 927314 599559 376395 454514 519230 057382 938410 866299 088049 110835 785349 757596 748036 > 4²³³ [i]