MinT - Best Known (41, 41+66, s)-Nets in Base 3

Best Known (41, 41+66, s)-Nets in Base 3

(41, 41+66, 42)-Net over F₃ — Constructive and digital

Digital (41, 107, 42)-net over F₃, using

t-expansion [i] based on digital (39, 107, 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]

(41, 41+66, 56)-Net over F₃ — Digital

Digital (41, 107, 56)-net over F₃, using

t-expansion [i] based on digital (40, 107, 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]

(41, 41+66, 198)-Net over F₃ — Upper bound on s (digital)

There is no digital (41, 107, 199)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁰⁷, 199, F₃, 66) (dual of [199, 92, 67]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹⁰⁶, 143, S₃, 66), but
    - the linear programming bound shows that MÂ â‰¥ 50584 630946 042848 451389 778465 029449 231952 748470 489372 922254 899547 818597 / 106 273880 037977 292025 > 3¹⁰⁶ [i]
  2. OA(3⁹², 199, 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]

(41, 41+66, 201)-Net in Base 3 — Upper bound on s

There is no (41, 107, 202)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1264 358854 277061 798307 648964 855637 873288 415448 448597 > 3¹⁰⁷ [i]