MinT - Best Known (129, 129+71, s)-Nets in Base 4

Best Known (129, 129+71, s)-Nets in Base 4

(129, 129+71, 157)-Net over F₄ — Constructive and digital

Digital (129, 200, 157)-net over F₄, using

4¹ times duplication [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]

(129, 129+71, 208)-Net in Base 4 — Constructive

(129, 200, 208)-net in base 4, using

trace code for nets [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]

(129, 129+71, 440)-Net over F₄ — Digital

Digital (129, 200, 440)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+71, 12253)-Net in Base 4 — Upper bound on s

There is no (129, 200, 12254)-net in base 4, because

1 times m-reduction [i] would yield (129, 199, 12254)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645985 590008 316988 199033 530014 177996 293932 470530 562649 481388 912672 245045 990776 720904 607058 862591 024398 168287 569032 863484 > 4¹⁹⁹ [i]