MinT - Best Known (53, 146, s)-Nets in Base 3

Best Known (53, 146, s)-Nets in Base 3

(53, 146, 48)-Net over F₃ — Constructive and digital

Digital (53, 146, 48)-net over F₃, using

t-expansion [i] based on digital (45, 146, 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]

(53, 146, 64)-Net over F₃ — Digital

Digital (53, 146, 64)-net over F₃, using

t-expansion [i] based on digital (49, 146, 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]

(53, 146, 229)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 146, 230)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁶, 230, F₃, 93) (dual of [230, 84, 94]-code), but
- residual code [i] would yield OA(3⁵³, 136, S₃, 31), but
  - the linear programming bound shows that MÂ â‰¥ 28 065209 106339 096583 292950 536466 552238 923456 737666 668230 316919 703911 / 1 370741 790957 279909 955012 441247 049995 659111 > 3⁵³ [i]

(53, 146, 244)-Net in Base 3 — Upper bound on s

There is no (53, 146, 245)-net in base 3, because

1 times m-reduction [i] would yield (53, 145, 245)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1661 304172 123144 797428 162585 224341 518515 741735 814134 431558 735871 095873 > 3¹⁴⁵ [i]