MinT - Best Known (181−67, 181, s)-Nets in Base 4

Best Known (181−67, 181, s)-Nets in Base 4

(181−67, 181, 140)-Net over F₄ — Constructive and digital

Digital (114, 181, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 35, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (79, 146, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 73, 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]

(181−67, 181, 152)-Net in Base 4 — Constructive

(114, 181, 152)-net in base 4, using

4¹ times duplication [i] based on (113, 180, 152)-net in base 4, using
- trace code for nets [i] based on (23, 90, 76)-net in base 16, using
  - base change [i] based on digital (5, 72, 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]

(181−67, 181, 349)-Net over F₄ — Digital

Digital (114, 181, 349)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(181−67, 181, 8410)-Net in Base 4 — Upper bound on s

There is no (114, 181, 8411)-net in base 4, because

1 times m-reduction [i] would yield (114, 180, 8411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 355936 866870 297000 362412 468607 615052 774329 995897 594538 640078 923113 106171 762475 231166 564656 339978 088844 221020 > 4¹⁸⁰ [i]