MinT - Best Known (184−64, 184, s)-Nets in Base 4

Best Known (184−64, 184, s)-Nets in Base 4

(184−64, 184, 158)-Net over F₄ — Constructive and digital

Digital (120, 184, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 44, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (76, 140, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 70, 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]

(184−64, 184, 208)-Net in Base 4 — Constructive

(120, 184, 208)-net in base 4, using

trace code for nets [i] based on (28, 92, 104)-net in base 16, using
- 3 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]

(184−64, 184, 439)-Net over F₄ — Digital

Digital (120, 184, 439)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(184−64, 184, 12322)-Net in Base 4 — Upper bound on s

There is no (120, 184, 12323)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 602 683526 925254 192914 969409 454914 930473 559591 102397 747291 197304 004169 563603 091541 646814 277182 905397 362253 481265 > 4¹⁸⁴ [i]