MinT - Best Known (210−72, 210, s)-Nets in Base 4

Best Known (210−72, 210, s)-Nets in Base 4

(210−72, 210, 163)-Net over F₄ — Constructive and digital

Digital (138, 210, 163)-net over F₄, using

2 times m-reduction [i] based on 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]

(210−72, 210, 240)-Net in Base 4 — Constructive

(138, 210, 240)-net in base 4, using

t-expansion [i] based on (137, 210, 240)-net in base 4, using
- trace code for nets [i] based on (32, 105, 120)-net in base 16, using
  - base change [i] based on digital (11, 84, 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]

(210−72, 210, 518)-Net over F₄ — Digital

Digital (138, 210, 518)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−72, 210, 15445)-Net in Base 4 — Upper bound on s

There is no (138, 210, 15446)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 710298 138769 231365 384748 188233 354736 027532 429603 180490 849387 509898 162862 210796 750256 558426 140282 774148 866245 018417 888823 550210 > 4²¹⁰ [i]