MinT - Best Known (166−105, 166, s)-Nets in Base 3

Best Known (166−105, 166, s)-Nets in Base 3

(166−105, 166, 48)-Net over F₃ — Constructive and digital

Digital (61, 166, 48)-net over F₃, using

t-expansion [i] based on digital (45, 166, 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]

(166−105, 166, 64)-Net over F₃ — Digital

Digital (61, 166, 64)-net over F₃, using

t-expansion [i] based on digital (49, 166, 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]

(166−105, 166, 265)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 166, 266)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁶⁶, 266, F₃, 105) (dual of [266, 100, 106]-code), but
- residual code [i] would yield OA(3⁶¹, 160, S₃, 35), but
  - the linear programming bound shows that MÂ â‰¥ 1093 910277 777742 998768 653454 514500 328731 948244 229348 565648 421721 114272 536195 / 8588 577766 921583 861591 295968 393699 452287 752003 > 3⁶¹ [i]

(166−105, 166, 281)-Net in Base 3 — Upper bound on s

There is no (61, 166, 282)-net in base 3, because

1 times m-reduction [i] would yield (61, 165, 282)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 468808 037737 109758 947061 092016 355176 250757 754925 596126 518279 210503 357215 138441 > 3¹⁶⁵ [i]