MinT - Best Known (126, 126+69, s)-Nets in Base 4

Best Known (126, 126+69, s)-Nets in Base 4

(126, 126+69, 157)-Net over F₄ — Constructive and digital

Digital (126, 195, 157)-net over F₄, using

4¹ times duplication [i] based on digital (125, 194, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 44, 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 (81, 150, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 75, 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]

(126, 126+69, 196)-Net in Base 4 — Constructive

(126, 195, 196)-net in base 4, using

3 times m-reduction [i] based on (126, 198, 196)-net in base 4, using
- trace code for nets [i] based on (27, 99, 98)-net in base 16, using
  - 1 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]

(126, 126+69, 435)-Net over F₄ — Digital

Digital (126, 195, 435)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+69, 12264)-Net in Base 4 — Upper bound on s

There is no (126, 195, 12265)-net in base 4, because

1 times m-reduction [i] would yield (126, 194, 12265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 631 098000 065749 573307 417834 907370 906182 937559 402190 981107 678530 068524 313722 516459 182107 401453 007887 746859 345501 213620 > 4¹⁹⁴ [i]