MinT - Best Known (178−118, 178, s)-Nets in Base 3

Best Known (178−118, 178, s)-Nets in Base 3

(178−118, 178, 48)-Net over F₃ — Constructive and digital

Digital (60, 178, 48)-net over F₃, using

t-expansion [i] based on digital (45, 178, 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]

(178−118, 178, 64)-Net over F₃ — Digital

Digital (60, 178, 64)-net over F₃, using

t-expansion [i] based on digital (49, 178, 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]

(178−118, 178, 197)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 178, 198)-net over F₃, because

1 times m-reduction [i] would yield digital (60, 177, 198)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁷, 198, F₃, 117) (dual of [198, 21, 118]-code), but
  - residual code [i] would yield OA(3⁶⁰, 80, S₃, 39), but
    - the linear programming bound shows that MÂ â‰¥ 10299 241947 274765 323111 769912 609288 895091 / 233101 383112 > 3⁶⁰ [i]

(178−118, 178, 259)-Net in Base 3 — Upper bound on s

There is no (60, 178, 260)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 293681 688662 419373 779212 403983 944971 846021 449652 766452 325613 385484 284720 421944 878705 > 3¹⁷⁸ [i]