MinT - Best Known (149, 149+81, s)-Nets in Base 4

Best Known (149, 149+81, s)-Nets in Base 4

(149, 149+81, 163)-Net over F₄ — Constructive and digital

Digital (149, 230, 163)-net over F₄, using

4¹ times duplication [i] based on digital (148, 229, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 55, 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 (93, 174, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 87, 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]

(149, 149+81, 240)-Net in Base 4 — Constructive

(149, 230, 240)-net in base 4, using

trace code for nets [i] based on (34, 115, 120)-net in base 16, using
- base change [i] based on digital (11, 92, 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]

(149, 149+81, 512)-Net over F₄ — Digital

Digital (149, 230, 512)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(149, 149+81, 14672)-Net in Base 4 — Upper bound on s

There is no (149, 230, 14673)-net in base 4, because

1 times m-reduction [i] would yield (149, 229, 14673)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 746032 156453 060806 504727 028777 467174 579942 185155 617184 334999 719373 318165 251224 290748 601019 932552 567736 789298 580089 464482 711740 862954 051332 > 4²²⁹ [i]