MinT - Best Known (198−70, 198, s)-Nets in Base 4

Best Known (198−70, 198, s)-Nets in Base 4

(198−70, 198, 157)-Net over F₄ — Constructive and digital

Digital (128, 198, 157)-net over F₄, using

1 times m-reduction [i] based on digital (128, 199, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 45, 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 (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(198−70, 198, 208)-Net in Base 4 — Constructive

(128, 198, 208)-net in base 4, using

trace code for nets [i] based on (29, 99, 104)-net in base 16, using
- 1 times m-reduction [i] based on (29, 100, 104)-net in base 16, using
  - base change [i] based on digital (9, 80, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(198−70, 198, 442)-Net over F₄ — Digital

Digital (128, 198, 442)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−70, 198, 11776)-Net in Base 4 — Upper bound on s

There is no (128, 198, 11777)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161489 762013 136718 868063 966714 440013 279302 167273 955462 694416 104131 064632 938640 461314 095877 810586 696252 684604 450163 297792 > 4¹⁹⁸ [i]