MinT - Best Known (112−69, 112, s)-Nets in Base 3

Best Known (112−69, 112, s)-Nets in Base 3

(112−69, 112, 42)-Net over F₃ — Constructive and digital

Digital (43, 112, 42)-net over F₃, using

t-expansion [i] based on digital (39, 112, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(112−69, 112, 56)-Net over F₃ — Digital

Digital (43, 112, 56)-net over F₃, using

t-expansion [i] based on digital (40, 112, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(112−69, 112, 203)-Net over F₃ — Upper bound on s (digital)

There is no digital (43, 112, 204)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹¹², 204, F₃, 69) (dual of [204, 92, 70]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹¹¹, 148, S₃, 69), but
    - the linear programming bound shows that MÂ â‰¥ 34269 689678 954827 771722 528323 125817 537099 985988 010230 663767 339699 553771 239936 / 338753 436767 266709 313125 > 3¹¹¹ [i]
  2. OA(3⁹², 204, S₃, 56), but
    - discarding factors would yield OA(3⁹², 150, S₃, 56), but
      - the linear programming bound shows that MÂ â‰¥ 380 530607 134695 199590 326105 296054 561976 304500 746203 230842 591578 312022 091639 435706 399387 387663 092379 / 4 623519 949937 639772 321756 933595 131069 999192 578955 859375 > 3⁹² [i]

(112−69, 112, 212)-Net in Base 3 — Upper bound on s

There is no (43, 112, 213)-net in base 3, because

1 times m-reduction [i] would yield (43, 111, 213)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 95370 079585 069187 638089 999215 079713 021577 050333 279217 > 3¹¹¹ [i]