MinT - Best Known (218−81, 218, s)-Nets in Base 4

Best Known (218−81, 218, s)-Nets in Base 4

(218−81, 218, 145)-Net over F₄ — Constructive and digital

Digital (137, 218, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 44, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (93, 174, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 87, 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]

(218−81, 218, 152)-Net in Base 4 — Constructive

(137, 218, 152)-net in base 4, using

2 times m-reduction [i] based on (137, 220, 152)-net in base 4, using
- trace code for nets [i] based on (27, 110, 76)-net in base 16, using
  - base change [i] based on digital (5, 88, 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]

(218−81, 218, 405)-Net over F₄ — Digital

Digital (137, 218, 405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−81, 218, 9668)-Net in Base 4 — Upper bound on s

There is no (137, 218, 9669)-net in base 4, because

1 times m-reduction [i] would yield (137, 217, 9669)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44389 613980 170746 363783 393300 662755 836605 437386 090532 667777 633218 525782 076201 288561 784255 480436 137427 806463 466111 343814 425210 656904 > 4²¹⁷ [i]