MinT - Best Known (56, 56+117, s)-Nets in Base 3

Best Known (56, 56+117, s)-Nets in Base 3

(56, 56+117, 48)-Net over F₃ — Constructive and digital

Digital (56, 173, 48)-net over F₃, using

t-expansion [i] based on digital (45, 173, 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]

(56, 56+117, 64)-Net over F₃ — Digital

Digital (56, 173, 64)-net over F₃, using

t-expansion [i] based on digital (49, 173, 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]

(56, 56+117, 177)-Net over F₃ — Upper bound on s (digital)

There is no digital (56, 173, 178)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷³, 178, F₃, 117) (dual of [178, 5, 118]-code), but
- Griesmer bound [i]

(56, 56+117, 238)-Net in Base 3 — Upper bound on s

There is no (56, 173, 239)-net in base 3, because

1 times m-reduction [i] would yield (56, 172, 239)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12616 335239 283312 211913 384123 389096 180107 254266 687968 412532 573871 733311 382759 263085 > 3¹⁷² [i]