MinT - Best Known (124, 124+71, s)-Nets in Base 3

Best Known (124, 124+71, s)-Nets in Base 3

(124, 124+71, 156)-Net over F₃ — Constructive and digital

Digital (124, 195, 156)-net over F₃, using

9 times m-reduction [i] based on digital (124, 204, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 102, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(124, 124+71, 239)-Net over F₃ — Digital

Digital (124, 195, 239)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+71, 3033)-Net in Base 3 — Upper bound on s

There is no (124, 195, 3034)-net in base 3, because

1 times m-reduction [i] would yield (124, 194, 3034)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 365 960298 357828 053574 374563 007943 049127 548102 321173 784369 913274 141982 003042 487299 250757 983025 > 3¹⁹⁴ [i]