MinT - Best Known (249−69, 249, s)-Nets in Base 3

Best Known (249−69, 249, s)-Nets in Base 3

(249−69, 249, 288)-Net over F₃ — Constructive and digital

Digital (180, 249, 288)-net over F₃, using

t-expansion [i] based on digital (177, 249, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 83, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(249−69, 249, 699)-Net over F₃ — Digital

Digital (180, 249, 699)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−69, 249, 20413)-Net in Base 3 — Upper bound on s

There is no (180, 249, 20414)-net in base 3, because

1 times m-reduction [i] would yield (180, 248, 20414)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21213 194704 864572 572796 187751 784933 830664 594008 227621 969142 623097 205788 716375 383545 912503 869595 765449 264506 881276 594365 > 3²⁴⁸ [i]