MinT - Best Known (164−57, 164, s)-Nets in Base 4

Best Known (164−57, 164, s)-Nets in Base 4

(164−57, 164, 157)-Net over F₄ — Constructive and digital

Digital (107, 164, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (69, 126, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 63, 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]

(164−57, 164, 196)-Net in Base 4 — Constructive

(107, 164, 196)-net in base 4, using

2 times m-reduction [i] based on (107, 166, 196)-net in base 4, using
- trace code for nets [i] based on (24, 83, 98)-net in base 16, using
  - 2 times m-reduction [i] based on (24, 85, 98)-net in base 16, using
    - base change [i] based on digital (7, 68, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(164−57, 164, 398)-Net over F₄ — Digital

Digital (107, 164, 398)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−57, 164, 12020)-Net in Base 4 — Upper bound on s

There is no (107, 164, 12021)-net in base 4, because

1 times m-reduction [i] would yield (107, 163, 12021)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 136 977786 158923 130806 836012 043157 184822 435127 103663 525580 876437 466480 627111 436018 054693 567625 377996 > 4¹⁶³ [i]