MinT - Best Known (177−117, 177, s)-Nets in Base 3

Best Known (177−117, 177, s)-Nets in Base 3

(177−117, 177, 48)-Net over F₃ — Constructive and digital

Digital (60, 177, 48)-net over F₃, using

t-expansion [i] based on digital (45, 177, 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]

(177−117, 177, 64)-Net over F₃ — Digital

Digital (60, 177, 64)-net over F₃, using

t-expansion [i] based on digital (49, 177, 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]

(177−117, 177, 197)-Net over F₃ — Upper bound on s (digital)

There is no digital (60, 177, 198)-net over F₃, because

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]

(177−117, 177, 261)-Net in Base 3 — Upper bound on s

There is no (60, 177, 262)-net in base 3, because

1 times m-reduction [i] would yield (60, 176, 262)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 077185 453534 314772 392316 484374 678165 729711 108657 076157 675543 115176 201981 177495 330653 > 3¹⁷⁶ [i]