MinT - Best Known (149−51, 149, s)-Nets in Base 4

Best Known (149−51, 149, s)-Nets in Base 4

(149−51, 149, 157)-Net over F₄ — Constructive and digital

Digital (98, 149, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 35, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (63, 114, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 57, 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]

(149−51, 149, 196)-Net in Base 4 — Constructive

(98, 149, 196)-net in base 4, using

t-expansion [i] based on (97, 149, 196)-net in base 4, using
- 1 times m-reduction [i] based on (97, 150, 196)-net in base 4, using
  - trace code for nets [i] based on (22, 75, 98)-net in base 16, using
    - base change [i] based on digital (7, 60, 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]

(149−51, 149, 388)-Net over F₄ — Digital

Digital (98, 149, 388)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(149−51, 149, 12416)-Net in Base 4 — Upper bound on s

There is no (98, 149, 12417)-net in base 4, because

1 times m-reduction [i] would yield (98, 148, 12417)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 127513 558545 183650 531691 152711 687564 321363 956304 055252 626079 401606 119039 031900 841161 182912 > 4¹⁴⁸ [i]