MinT - Best Known (53, 53+95, s)-Nets in Base 3

Best Known (53, 53+95, s)-Nets in Base 3

(53, 53+95, 48)-Net over F₃ — Constructive and digital

Digital (53, 148, 48)-net over F₃, using

t-expansion [i] based on digital (45, 148, 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, 53+95, 64)-Net over F₃ — Digital

Digital (53, 148, 64)-net over F₃, using

t-expansion [i] based on digital (49, 148, 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, 53+95, 229)-Net over F₃ — Upper bound on s (digital)

There is no digital (53, 148, 230)-net over F₃, because

2 times m-reduction [i] would yield digital (53, 146, 230)-net over F₃, but
- 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, 53+95, 241)-Net in Base 3 — Upper bound on s

There is no (53, 148, 242)-net in base 3, because

1 times m-reduction [i] would yield (53, 147, 242)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14327 002633 278780 843079 099431 324415 923643 085972 965564 539260 147854 011033 > 3¹⁴⁷ [i]