MinT - Best Known (193−127, 193, s)-Nets in Base 3

Best Known (193−127, 193, s)-Nets in Base 3

(193−127, 193, 48)-Net over F₃ — Constructive and digital

Digital (66, 193, 48)-net over F₃, using

t-expansion [i] based on digital (45, 193, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(193−127, 193, 64)-Net over F₃ — Digital

Digital (66, 193, 64)-net over F₃, using

t-expansion [i] based on digital (49, 193, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(193−127, 193, 220)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 193, 221)-net over F₃, because

1 times m-reduction [i] would yield digital (66, 192, 221)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹², 221, F₃, 126) (dual of [221, 29, 127]-code), but
  - residual code [i] would yield OA(3⁶⁶, 94, S₃, 42), but
    - the linear programming bound shows that MÂ â‰¥ 1 747070 674352 666159 830908 506217 598892 875513 970337 / 51301 241878 906250 > 3⁶⁶ [i]

(193−127, 193, 287)-Net in Base 3 — Upper bound on s

There is no (66, 193, 288)-net in base 3, because

1 times m-reduction [i] would yield (66, 192, 288)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 44 301723 955894 233090 669483 358257 932234 573529 660774 264990 376749 853736 427456 686432 367594 171777 > 3¹⁹² [i]