MinT - Best Known (127, 127+63, s)-Nets in Base 3

Best Known (127, 127+63, s)-Nets in Base 3

(127, 127+63, 162)-Net over F₃ — Constructive and digital

Digital (127, 190, 162)-net over F₃, using

trace code for nets [i] based on digital (32, 95, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(127, 127+63, 310)-Net over F₃ — Digital

Digital (127, 190, 310)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+63, 5003)-Net in Base 3 — Upper bound on s

There is no (127, 190, 5004)-net in base 3, because

1 times m-reduction [i] would yield (127, 189, 5004)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 504350 407718 759548 825240 177355 630743 437355 945881 675298 826385 290623 642545 127281 617182 918161 > 3¹⁸⁹ [i]