MinT - Best Known (249−63, 249, s)-Nets in Base 3

Best Known (249−63, 249, s)-Nets in Base 3

(249−63, 249, 324)-Net over F₃ — Constructive and digital

Digital (186, 249, 324)-net over F₃, using

t-expansion [i] based on digital (184, 249, 324)-net over F₃, using
- trace code for nets [i] based on digital (18, 83, 108)-net over F₂₇, using
  - net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
      - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(249−63, 249, 974)-Net over F₃ — Digital

Digital (186, 249, 974)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−63, 249, 40705)-Net in Base 3 — Upper bound on s

There is no (186, 249, 40706)-net in base 3, because

1 times m-reduction [i] would yield (186, 248, 40706)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21203 132776 637044 181984 544232 462718 939453 202760 406126 953524 800107 508563 619610 156493 570999 061347 331995 258542 482262 309625 > 3²⁴⁸ [i]