MinT - Best Known (88−31, 88, s)-Nets in Base 3

Best Known (88−31, 88, s)-Nets in Base 3

(88−31, 88, 128)-Net over F₃ — Constructive and digital

Digital (57, 88, 128)-net over F₃, using

trace code for nets [i] based on digital (13, 44, 64)-net over F₉, using
- net from sequence [i] based on digital (13, 63)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
    - a function field by GarcÃa and Quoos [i]

(88−31, 88, 138)-Net over F₃ — Digital

Digital (57, 88, 138)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−31, 88, 1865)-Net in Base 3 — Upper bound on s

There is no (57, 88, 1866)-net in base 3, because

1 times m-reduction [i] would yield (57, 87, 1866)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 325753 270028 629711 096102 353414 785740 636665 > 3⁸⁷ [i]