MinT - Best Known (65, 65+128, s)-Nets in Base 3

Best Known (65, 65+128, s)-Nets in Base 3

(65, 65+128, 48)-Net over F₃ — Constructive and digital

Digital (65, 193, 48)-net over F₃, using

t-expansion [i] based on digital (45, 193, 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]

(65, 65+128, 64)-Net over F₃ — Digital

Digital (65, 193, 64)-net over F₃, using

t-expansion [i] based on digital (49, 193, 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]

(65, 65+128, 212)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 193, 213)-net over F₃, because

2 times m-reduction [i] would yield digital (65, 191, 213)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁹¹, 213, F₃, 126) (dual of [213, 22, 127]-code), but
  - residual code [i] would yield OA(3⁶⁵, 86, S₃, 42), but
    - the linear programming bound shows that MÂ â‰¥ 105639 517822 541973 415942 878050 095499 770611 / 8844 615395 > 3⁶⁵ [i]

(65, 65+128, 279)-Net in Base 3 — Upper bound on s

There is no (65, 193, 280)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 124 458763 118644 103172 363406 614703 275564 300010 868244 167907 665939 877508 547927 160814 663009 538049 > 3¹⁹³ [i]