MinT - Best Known (186−122, 186, s)-Nets in Base 3

Best Known (186−122, 186, s)-Nets in Base 3

(186−122, 186, 48)-Net over F₃ — Constructive and digital

Digital (64, 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]

(186−122, 186, 64)-Net over F₃ — Digital

Digital (64, 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]

(186−122, 186, 224)-Net over F₃ — Upper bound on s (digital)

There is no digital (64, 186, 225)-net over F₃, because

2 times m-reduction [i] would yield digital (64, 184, 225)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁸⁴, 225, F₃, 120) (dual of [225, 41, 121]-code), but
  - residual code [i] would yield OA(3⁶⁴, 104, S₃, 40), but
    - the linear programming bound shows that MÂ â‰¥ 2 962004 263094 532507 337869 081743 858222 059094 257308 728871 435455 157099 / 842061 707484 948215 471641 634707 432925 > 3⁶⁴ [i]

(186−122, 186, 279)-Net in Base 3 — Upper bound on s

There is no (64, 186, 280)-net in base 3, because

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]