MinT - Best Known (189−125, 189, s)-Nets in Base 3

Best Known (189−125, 189, s)-Nets in Base 3

(189−125, 189, 48)-Net over F₃ — Constructive and digital

Digital (64, 189, 48)-net over F₃, using

t-expansion [i] based on digital (45, 189, 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]

(189−125, 189, 64)-Net over F₃ — Digital

Digital (64, 189, 64)-net over F₃, using

t-expansion [i] based on digital (49, 189, 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]

(189−125, 189, 212)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 189, 213)-net over F₃, because

2 times m-reduction [i] would yield digital (64, 187, 213)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁷, 213, F₃, 123) (dual of [213, 26, 124]-code), but
  - residual code [i] would yield OA(3⁶⁴, 89, S₃, 41), but
    - the linear programming bound shows that MÂ â‰¥ 270 344305 793094 767616 866429 490879 306189 053619 / 74 160889 643750 > 3⁶⁴ [i]

(189−125, 189, 277)-Net in Base 3 — Upper bound on s

There is no (64, 189, 278)-net in base 3, because

1 times m-reduction [i] would yield (64, 188, 278)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 545056 173696 044863 632412 413956 013766 867186 166786 867654 891463 473676 713648 026753 392186 263973 > 3¹⁸⁸ [i]