MinT - Best Known (115, 180, s)-Nets in Base 4

Best Known (115, 180, s)-Nets in Base 4

(115, 180, 147)-Net over F₄ — Constructive and digital

Digital (115, 180, 147)-net over F₄, using

4¹ times duplication [i] based on digital (114, 179, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 37, 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 (77, 142, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 71, 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]

(115, 180, 196)-Net in Base 4 — Constructive

(115, 180, 196)-net in base 4, using

trace code for nets [i] based on (25, 90, 98)-net in base 16, using
- base change [i] based on digital (7, 72, 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]

(115, 180, 378)-Net over F₄ — Digital

Digital (115, 180, 378)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 180, 9917)-Net in Base 4 — Upper bound on s

There is no (115, 180, 9918)-net in base 4, because

1 times m-reduction [i] would yield (115, 179, 9918)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 588654 014916 717922 367360 531511 094877 900960 855240 008554 687023 630911 476768 095281 340705 132110 948060 722912 303703 > 4¹⁷⁹ [i]