MinT - Best Known (184−63, 184, s)-Nets in Base 4

Best Known (184−63, 184, s)-Nets in Base 4

(184−63, 184, 163)-Net over F₄ — Constructive and digital

Digital (121, 184, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 46, 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 (75, 138, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 69, 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]

(184−63, 184, 208)-Net in Base 4 — Constructive

(121, 184, 208)-net in base 4, using

2 times m-reduction [i] based on (121, 186, 208)-net in base 4, using
- trace code for nets [i] based on (28, 93, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (28, 95, 104)-net in base 16, using
    - base change [i] based on digital (9, 76, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(184−63, 184, 464)-Net over F₄ — Digital

Digital (121, 184, 464)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(184−63, 184, 14800)-Net in Base 4 — Upper bound on s

There is no (121, 184, 14801)-net in base 4, because

1 times m-reduction [i] would yield (121, 183, 14801)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 580450 360351 395913 143448 419476 011203 985883 320468 751497 335286 558751 131542 686452 012073 301408 468723 850327 096000 > 4¹⁸³ [i]