MinT - Best Known (149−52, 149, s)-Nets in Base 4

Best Known (149−52, 149, s)-Nets in Base 4

(149−52, 149, 151)-Net over F₄ — Constructive and digital

Digital (97, 149, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 33, 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 (64, 116, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 58, 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]

(149−52, 149, 196)-Net in Base 4 — Constructive

(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]

(149−52, 149, 362)-Net over F₄ — Digital

Digital (97, 149, 362)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(149−52, 149, 9897)-Net in Base 4 — Upper bound on s

There is no (97, 149, 9898)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 510350 670362 083775 780403 559244 723344 242058 102796 784261 916908 519650 818684 470168 527298 803680 > 4¹⁴⁹ [i]