MinT - Best Known (51, 51+107, s)-Nets in Base 3

Best Known (51, 51+107, s)-Nets in Base 3

(51, 51+107, 48)-Net over F₃ — Constructive and digital

Digital (51, 158, 48)-net over F₃, using

t-expansion [i] based on digital (45, 158, 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]

(51, 51+107, 64)-Net over F₃ — Digital

Digital (51, 158, 64)-net over F₃, using

t-expansion [i] based on digital (49, 158, 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]

(51, 51+107, 163)-Net over F₃ — Upper bound on s (digital)

There is no digital (51, 158, 164)-net over F₃, because

2 times m-reduction [i] would yield digital (51, 156, 164)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁶, 164, F₃, 105) (dual of [164, 8, 106]-code), but
  - residual code [i] would yield OA(3⁵¹, 58, S₃, 35), but
    - the linear programming bound shows that MÂ â‰¥ 2151 540269 112482 208544 436253 / 946 > 3⁵¹ [i]

(51, 51+107, 218)-Net in Base 3 — Upper bound on s

There is no (51, 158, 219)-net in base 3, because

1 times m-reduction [i] would yield (51, 157, 219)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 958 541223 003097 307290 969328 247174 794778 692282 559442 638377 682496 867373 758095 > 3¹⁵⁷ [i]