MinT - Best Known (243−119, 243, s)-Nets in Base 3

Best Known (243−119, 243, s)-Nets in Base 3

(243−119, 243, 78)-Net over F₃ — Constructive and digital

Digital (124, 243, 78)-net over F₃, using

t-expansion [i] based on digital (121, 243, 78)-net over F₃, using
- net from sequence [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [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]

(243−119, 243, 126)-Net over F₃ — Digital

Digital (124, 243, 126)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−119, 243, 976)-Net in Base 3 — Upper bound on s

There is no (124, 243, 977)-net in base 3, because

1 times m-reduction [i] would yield (124, 242, 977)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30 163018 913398 275587 721299 979296 943544 807106 031123 689356 537907 356893 338173 532307 858099 587428 716054 440338 306725 979131 > 3²⁴² [i]