MinT - Best Known (172−114, 172, s)-Nets in Base 3

Best Known (172−114, 172, s)-Nets in Base 3

(172−114, 172, 48)-Net over F₃ — Constructive and digital

Digital (58, 172, 48)-net over F₃, using

t-expansion [i] based on digital (45, 172, 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]

(172−114, 172, 64)-Net over F₃ — Digital

Digital (58, 172, 64)-net over F₃, using

t-expansion [i] based on digital (49, 172, 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]

(172−114, 172, 189)-Net over F₃ — Upper bound on s (digital)

There is no digital (58, 172, 190)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷², 190, F₃, 114) (dual of [190, 18, 115]-code), but
- residual code [i] would yield OA(3⁵⁸, 75, S₃, 38), but
  - the linear programming bound shows that MÂ â‰¥ 425880 948792 153754 309607 908493 218857 / 85 592416 > 3⁵⁸ [i]

(172−114, 172, 251)-Net in Base 3 — Upper bound on s

There is no (58, 172, 252)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13109 403647 354297 299322 935511 196512 926659 236947 567561 012760 780653 305964 500532 776057 > 3¹⁷² [i]