MinT - Best Known (143, 143+85, s)-Nets in Base 4

Best Known (143, 143+85, s)-Nets in Base 4

(143, 143+85, 145)-Net over F₄ — Constructive and digital

Digital (143, 228, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 46, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, 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]

(143, 143+85, 152)-Net in Base 4 — Constructive

(143, 228, 152)-net in base 4, using

2 times m-reduction [i] based on (143, 230, 152)-net in base 4, using
- trace code for nets [i] based on (28, 115, 76)-net in base 16, using
  - base change [i] based on digital (5, 92, 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]

(143, 143+85, 416)-Net over F₄ — Digital

Digital (143, 228, 416)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(143, 143+85, 9843)-Net in Base 4 — Upper bound on s

There is no (143, 228, 9844)-net in base 4, because

1 times m-reduction [i] would yield (143, 227, 9844)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46596 313702 757045 742931 140456 931547 572328 101757 945361 641274 240402 949306 549891 532114 182527 579450 611132 918975 182891 463989 182183 204775 597690 > 4²²⁷ [i]