MinT - Best Known (127, 200, s)-Nets in Base 4

Best Known (127, 200, s)-Nets in Base 4

(127, 200, 147)-Net over F₄ — Constructive and digital

Digital (127, 200, 147)-net over F₄, using

4¹ times duplication [i] based on digital (126, 199, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 41, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (85, 158, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 79, 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]

(127, 200, 196)-Net in Base 4 — Constructive

(127, 200, 196)-net in base 4, using

trace code for nets [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]

(127, 200, 400)-Net over F₄ — Digital

Digital (127, 200, 400)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 200, 10101)-Net in Base 4 — Upper bound on s

There is no (127, 200, 10102)-net in base 4, because

1 times m-reduction [i] would yield (127, 199, 10102)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645705 058820 969633 529037 422221 628924 632916 626899 348492 482375 702217 351388 461758 950628 985407 200682 287651 380704 476000 095297 > 4¹⁹⁹ [i]