MinT - Best Known (165−104, 165, s)-Nets in Base 3

Best Known (165−104, 165, s)-Nets in Base 3

(165−104, 165, 48)-Net over F₃ — Constructive and digital

Digital (61, 165, 48)-net over F₃, using

t-expansion [i] based on digital (45, 165, 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]

(165−104, 165, 64)-Net over F₃ — Digital

Digital (61, 165, 64)-net over F₃, using

t-expansion [i] based on digital (49, 165, 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]

(165−104, 165, 278)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 165, 279)-net over F₃, because

2 times m-reduction [i] would yield digital (61, 163, 279)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁶³, 279, F₃, 102) (dual of [279, 116, 103]-code), but
  - residual code [i] would yield OA(3⁶¹, 176, S₃, 34), but
    - the linear programming bound shows that MÂ â‰¥ 812 356626 791711 438639 003017 318488 339021 708011 289038 629210 828125 / 5801 809430 490682 982977 100374 687357 > 3⁶¹ [i]

(165−104, 165, 281)-Net in Base 3 — Upper bound on s

There is no (61, 165, 282)-net in base 3, because

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]