MinT - Best Known (187−123, 187, s)-Nets in Base 3

Best Known (187−123, 187, s)-Nets in Base 3

(187−123, 187, 48)-Net over F₃ — Constructive and digital

Digital (64, 187, 48)-net over F₃, using

t-expansion [i] based on digital (45, 187, 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]

(187−123, 187, 64)-Net over F₃ — Digital

Digital (64, 187, 64)-net over F₃, using

t-expansion [i] based on digital (49, 187, 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]

(187−123, 187, 212)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 187, 213)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁷, 213, F₃, 123) (dual of [213, 26, 124]-code), but
- residual code [i] would yield OA(3⁶⁴, 89, S₃, 41), but
  - the linear programming bound shows that MÂ â‰¥ 270 344305 793094 767616 866429 490879 306189 053619 / 74 160889 643750 > 3⁶⁴ [i]

(187−123, 187, 279)-Net in Base 3 — Upper bound on s

There is no (64, 187, 280)-net in base 3, because

1 times m-reduction [i] would yield (64, 186, 280)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 62058 617562 558324 550439 166616 749645 988818 835559 908566 077386 358647 993985 974532 816152 071729 > 3¹⁸⁶ [i]