MinT - Best Known (65, 65+123, s)-Nets in Base 3

Best Known (65, 65+123, s)-Nets in Base 3

(65, 65+123, 48)-Net over F₃ — Constructive and digital

Digital (65, 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]

(65, 65+123, 64)-Net over F₃ — Digital

Digital (65, 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]

(65, 65+123, 222)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 188, 223)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁸, 223, F₃, 123) (dual of [223, 35, 124]-code), but
- residual code [i] would yield OA(3⁶⁵, 99, S₃, 41), but
  - the linear programming bound shows that MÂ â‰¥ 1 398289 528640 015865 013744 903134 488973 245855 682309 745191 / 125126 745302 260144 143218 > 3⁶⁵ [i]

(65, 65+123, 285)-Net in Base 3 — Upper bound on s

There is no (65, 188, 286)-net in base 3, because

1 times m-reduction [i] would yield (65, 187, 286)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 184668 308290 415647 471079 783218 815561 131166 287257 139761 135080 542735 281964 873882 765419 684581 > 3¹⁸⁷ [i]