MinT - Best Known (174−62, 174, s)-Nets in Base 4

Best Known (174−62, 174, s)-Nets in Base 4

(174−62, 174, 151)-Net over F₄ — Constructive and digital

Digital (112, 174, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 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 (74, 136, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 68, 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]

(174−62, 174, 196)-Net in Base 4 — Constructive

(112, 174, 196)-net in base 4, using

trace code for nets [i] based on (25, 87, 98)-net in base 16, using
- 3 times m-reduction [i] based on (25, 90, 98)-net in base 16, using
  - base change [i] based on digital (7, 72, 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]

(174−62, 174, 384)-Net over F₄ — Digital

Digital (112, 174, 384)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−62, 174, 9887)-Net in Base 4 — Upper bound on s

There is no (112, 174, 9888)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 573 474847 440232 315863 213363 165701 700455 565770 098899 343291 165227 371784 959524 450274 607353 001687 854419 287895 > 4¹⁷⁴ [i]