MinT - Best Known (133, 133+73, s)-Nets in Base 4

Best Known (133, 133+73, s)-Nets in Base 4

(133, 133+73, 158)-Net over F₄ — Constructive and digital

Digital (133, 206, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 48, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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]

(133, 133+73, 208)-Net in Base 4 — Constructive

(133, 206, 208)-net in base 4, using

trace code for nets [i] based on (30, 103, 104)-net in base 16, using
- 2 times m-reduction [i] based on (30, 105, 104)-net in base 16, using
  - base change [i] based on digital (9, 84, 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]

(133, 133+73, 454)-Net over F₄ — Digital

Digital (133, 206, 454)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+73, 12735)-Net in Base 4 — Upper bound on s

There is no (133, 206, 12736)-net in base 4, because

1 times m-reduction [i] would yield (133, 205, 12736)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2649 475438 115046 536545 437763 804369 945156 363458 512895 054501 580837 030006 055652 640942 037812 163739 438483 853354 276529 048121 703567 > 4²⁰⁵ [i]