MinT - Best Known (235−87, 235, s)-Nets in Base 4

Best Known (235−87, 235, s)-Nets in Base 4

(235−87, 235, 147)-Net over F₄ — Constructive and digital

Digital (148, 235, 147)-net over F₄, using

4¹ times duplication [i] based on digital (147, 234, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 48, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (99, 186, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 93, 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]

(235−87, 235, 152)-Net in Base 4 — Constructive

(148, 235, 152)-net in base 4, using

3 times m-reduction [i] based on (148, 238, 152)-net in base 4, using
- trace code for nets [i] based on (29, 119, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (29, 120, 76)-net in base 16, using
    - base change [i] based on digital (5, 96, 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]

(235−87, 235, 438)-Net over F₄ — Digital

Digital (148, 235, 438)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−87, 235, 10598)-Net in Base 4 — Upper bound on s

There is no (148, 235, 10599)-net in base 4, because

1 times m-reduction [i] would yield (148, 234, 10599)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 764 768530 070869 946471 866087 550868 151280 130608 180049 736390 527425 160356 387595 247805 883781 116889 042525 957729 042535 984591 971909 144197 837987 739456 > 4²³⁴ [i]