MinT - Best Known (164−103, 164, s)-Nets in Base 3

Best Known (164−103, 164, s)-Nets in Base 3

(164−103, 164, 48)-Net over F₃ — Constructive and digital

Digital (61, 164, 48)-net over F₃, using

t-expansion [i] based on digital (45, 164, 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]

(164−103, 164, 64)-Net over F₃ — Digital

Digital (61, 164, 64)-net over F₃, using

t-expansion [i] based on digital (49, 164, 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]

(164−103, 164, 278)-Net over F₃ — Upper bound on s (digital)

There is no digital (61, 164, 279)-net over F₃, because

1 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]

(164−103, 164, 284)-Net in Base 3 — Upper bound on s

There is no (61, 164, 285)-net in base 3, because

1 times m-reduction [i] would yield (61, 163, 285)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 597386 814336 722108 071472 330466 460642 155298 705715 315669 156361 152274 558927 242571 > 3¹⁶³ [i]