MinT - Best Known (230−85, 230, s)-Nets in Base 4

Best Known (230−85, 230, s)-Nets in Base 4

(230−85, 230, 147)-Net over F₄ — Constructive and digital

Digital (145, 230, 147)-net over F₄, using

4¹ times duplication [i] based on digital (144, 229, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 47, 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 (97, 182, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 91, 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]

(230−85, 230, 196)-Net in Base 4 — Constructive

(145, 230, 196)-net in base 4, using

trace code for nets [i] based on (30, 115, 98)-net in base 16, using
- base change [i] based on digital (7, 92, 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]

(230−85, 230, 432)-Net over F₄ — Digital

Digital (145, 230, 432)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−85, 230, 10517)-Net in Base 4 — Upper bound on s

There is no (145, 230, 10518)-net in base 4, because

1 times m-reduction [i] would yield (145, 229, 10518)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 745186 629506 105764 385539 412301 090123 661405 637817 885630 959152 160978 389133 183863 944054 040207 921892 387072 739595 864904 708425 444391 045305 018144 > 4²²⁹ [i]