MinT - Best Known (125, 125+71, s)-Nets in Base 4

Best Known (125, 125+71, s)-Nets in Base 4

(125, 125+71, 151)-Net over F₄ — Constructive and digital

Digital (125, 196, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 42, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (83, 154, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 77, 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]

(125, 125+71, 196)-Net in Base 4 — Constructive

(125, 196, 196)-net in base 4, using

trace code for nets [i] based on (27, 98, 98)-net in base 16, using
- 2 times m-reduction [i] based on (27, 100, 98)-net in base 16, using
  - base change [i] based on digital (7, 80, 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]

(125, 125+71, 404)-Net over F₄ — Digital

Digital (125, 196, 404)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+71, 10454)-Net in Base 4 — Upper bound on s

There is no (125, 196, 10455)-net in base 4, because

1 times m-reduction [i] would yield (125, 195, 10455)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2528 889423 907318 708356 998559 666998 153859 090757 093592 506682 076371 514504 540773 502040 198514 192856 446627 727532 310064 793408 > 4¹⁹⁵ [i]