MinT - Best Known (144−51, 144, s)-Nets in Base 4

Best Known (144−51, 144, s)-Nets in Base 4

(144−51, 144, 147)-Net over F₄ — Constructive and digital

Digital (93, 144, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 30, 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 (63, 114, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 57, 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]

(144−51, 144, 152)-Net in Base 4 — Constructive

(93, 144, 152)-net in base 4, using

2 times m-reduction [i] based on (93, 146, 152)-net in base 4, using
- trace code for nets [i] based on (20, 73, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (20, 75, 76)-net in base 16, using
    - base change [i] based on digital (5, 60, 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]

(144−51, 144, 334)-Net over F₄ — Digital

Digital (93, 144, 334)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−51, 144, 9404)-Net in Base 4 — Upper bound on s

There is no (93, 144, 9405)-net in base 4, because

1 times m-reduction [i] would yield (93, 143, 9405)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 124 389635 350524 604692 280267 148936 630512 633414 908759 885874 795288 327656 502001 508350 873392 > 4¹⁴³ [i]