MinT - Best Known (212−74, 212, s)-Nets in Base 4

Best Known (212−74, 212, s)-Nets in Base 4

(212−74, 212, 163)-Net over F₄ — Constructive and digital

Digital (138, 212, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 52, 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 (86, 160, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 80, 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]

(212−74, 212, 208)-Net in Base 4 — Constructive

(138, 212, 208)-net in base 4, using

2 times m-reduction [i] based on (138, 214, 208)-net in base 4, using
- trace code for nets [i] based on (31, 107, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
    - base change [i] based on digital (9, 88, 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]

(212−74, 212, 491)-Net over F₄ — Digital

Digital (138, 212, 491)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−74, 212, 13723)-Net in Base 4 — Upper bound on s

There is no (138, 212, 13724)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 349368 438589 250217 046937 249749 430146 817562 999712 551883 270811 392000 934141 550384 778611 297353 634144 873622 874068 323201 327339 980520 > 4²¹² [i]