MinT - Best Known (194−70, 194, s)-Nets in Base 4

Best Known (194−70, 194, s)-Nets in Base 4

(194−70, 194, 151)-Net over F₄ — Constructive and digital

Digital (124, 194, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 42, 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 (82, 152, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 76, 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]

(194−70, 194, 196)-Net in Base 4 — Constructive

(124, 194, 196)-net in base 4, using

trace code for nets [i] based on (27, 97, 98)-net in base 16, using
- 3 times m-reduction [i] based on (27, 100, 98)-net in base 16, using
  - base change [i] based on digital (7, 80, 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]

(194−70, 194, 405)-Net over F₄ — Digital

Digital (124, 194, 405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−70, 194, 10047)-Net in Base 4 — Upper bound on s

There is no (124, 194, 10048)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 632 492460 273313 891735 425100 986080 430413 378672 251065 026882 793931 127373 196351 383604 212381 048110 161811 158140 764666 969947 > 4¹⁹⁴ [i]