MinT - Best Known (184−66, 184, s)-Nets in Base 4

Best Known (184−66, 184, s)-Nets in Base 4

(184−66, 184, 151)-Net over F₄ — Constructive and digital

Digital (118, 184, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 40, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (78, 144, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 72, 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]

(184−66, 184, 196)-Net in Base 4 — Constructive

(118, 184, 196)-net in base 4, using

trace code for nets [i] based on (26, 92, 98)-net in base 16, using
- 3 times m-reduction [i] based on (26, 95, 98)-net in base 16, using
  - base change [i] based on digital (7, 76, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(184−66, 184, 395)-Net over F₄ — Digital

Digital (118, 184, 395)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(184−66, 184, 9954)-Net in Base 4 — Upper bound on s

There is no (118, 184, 9955)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 603 191434 144982 398448 456493 696038 133558 231718 850081 118699 666307 325196 717183 893031 645596 960554 330211 084282 620942 > 4¹⁸⁴ [i]