MinT - Best Known (63, 188, s)-Nets in Base 3

Best Known (63, 188, s)-Nets in Base 3

(63, 188, 48)-Net over F₃ — Constructive and digital

Digital (63, 188, 48)-net over F₃, using

t-expansion [i] based on digital (45, 188, 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, 188, 64)-Net over F₃ — Digital

Digital (63, 188, 64)-net over F₃, using

t-expansion [i] based on digital (49, 188, 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, 188, 204)-Net over F₃ — Upper bound on s (digital)

There is no digital (63, 188, 205)-net over F₃, because

2 times m-reduction [i] would yield digital (63, 186, 205)-net over F₃, but
- 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, 188, 271)-Net in Base 3 — Upper bound on s

There is no (63, 188, 272)-net in base 3, because

1 times m-reduction [i] would yield (63, 187, 272)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 175482 905175 262769 591113 186283 445566 187114 188278 373867 486894 140344 786746 898561 926028 753249 > 3¹⁸⁷ [i]