MinT - Best Known (63, 63+123, s)-Nets in Base 3

Best Known (63, 63+123, s)-Nets in Base 3

(63, 63+123, 48)-Net over F₃ — Constructive and digital

Digital (63, 186, 48)-net over F₃, using

t-expansion [i] based on digital (45, 186, 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]

(63, 63+123, 64)-Net over F₃ — Digital

Digital (63, 186, 64)-net over F₃, using

t-expansion [i] based on digital (49, 186, 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]

(63, 63+123, 204)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 186, 205)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁶, 205, F₃, 123) (dual of [205, 19, 124]-code), but
- residual code [i] would yield OA(3⁶³, 81, S₃, 41), but
  - the linear programming bound shows that MÂ â‰¥ 645814 120288 256333 139417 572410 149042 / 494527 > 3⁶³ [i]

(63, 63+123, 273)-Net in Base 3 — Upper bound on s

There is no (63, 186, 274)-net in base 3, because

1 times m-reduction [i] would yield (63, 185, 274)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 20436 524157 726527 138275 743747 433931 685184 602066 959129 536675 907915 980417 458559 835917 614861 > 3¹⁸⁵ [i]