MinT - Best Known (178−117, 178, s)-Nets in Base 3

Best Known (178−117, 178, s)-Nets in Base 3

(178−117, 178, 48)-Net over F₃ — Constructive and digital

Digital (61, 178, 48)-net over F₃, using

t-expansion [i] based on digital (45, 178, 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]

(178−117, 178, 64)-Net over F₃ — Digital

Digital (61, 178, 64)-net over F₃, using

t-expansion [i] based on digital (49, 178, 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]

(178−117, 178, 205)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 178, 206)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁸, 206, F₃, 117) (dual of [206, 28, 118]-code), but
- residual code [i] would yield OA(3⁶¹, 88, S₃, 39), but
  - the linear programming bound shows that MÂ â‰¥ 27794 948374 262252 134892 330586 188426 701201 134663 / 193694 611824 634375 > 3⁶¹ [i]

(178−117, 178, 267)-Net in Base 3 — Upper bound on s

There is no (61, 178, 268)-net in base 3, because

1 times m-reduction [i] would yield (61, 177, 268)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 252222 167550 503418 418526 545197 672077 514365 669236 889033 123944 528727 586431 732553 912681 > 3¹⁷⁷ [i]