MinT - Best Known (126−59, 126, s)-Nets in Base 3

Best Known (126−59, 126, s)-Nets in Base 3

(126−59, 126, 60)-Net over F₃ — Constructive and digital

Digital (67, 126, 60)-net over F₃, using

trace code for nets [i] based on digital (4, 63, 30)-net over F₉, using
- net from sequence [i] based on digital (4, 29)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
    - a function field by SÃ©mirat [i]

(126−59, 126, 82)-Net over F₃ — Digital

Digital (67, 126, 82)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−59, 126, 636)-Net in Base 3 — Upper bound on s

There is no (67, 126, 637)-net in base 3, because

1 times m-reduction [i] would yield (67, 125, 637)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 441758 471684 931830 719598 003003 468550 260479 669298 430469 399851 > 3¹²⁵ [i]