MinT - Best Known (218−75, 218, s)-Nets in Base 4

Best Known (218−75, 218, s)-Nets in Base 4

(218−75, 218, 163)-Net over F₄ — Constructive and digital

Digital (143, 218, 163)-net over F₄, using

t-expansion [i] based on digital (142, 218, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (142, 219, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 53, 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 (89, 166, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 83, 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]

(218−75, 218, 240)-Net in Base 4 — Constructive

(143, 218, 240)-net in base 4, using

2 times m-reduction [i] based on (143, 220, 240)-net in base 4, using
- trace code for nets [i] based on (33, 110, 120)-net in base 16, using
  - base change [i] based on digital (11, 88, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(218−75, 218, 529)-Net over F₄ — Digital

Digital (143, 218, 529)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−75, 218, 16557)-Net in Base 4 — Upper bound on s

There is no (143, 218, 16558)-net in base 4, because

1 times m-reduction [i] would yield (143, 217, 16558)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44404 894680 236393 030853 650187 229860 960058 426113 762373 803913 624342 099252 498947 182817 145276 479309 737390 259841 161498 045922 695065 290585 > 4²¹⁷ [i]