MinT - Best Known (216−37, 216, s)-Nets in Base 3

Best Known (216−37, 216, s)-Nets in Base 3

(216−37, 216, 1480)-Net over F₃ — Constructive and digital

Digital (179, 216, 1480)-net over F₃, using

t-expansion [i] based on digital (178, 216, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(216−37, 216, 5897)-Net over F₃ — Digital

Digital (179, 216, 5897)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁶, 5897, F₃, 37) (dual of [5897, 5681, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁶, 9841, F₃, 37) (dual of [9841, 9625, 38]-code), using
  - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,782,1563,…,−1406}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]

(216−37, 216, 1888184)-Net in Base 3 — Upper bound on s

There is no (179, 216, 1888185)-net in base 3, because

1 times m-reduction [i] would yield (179, 215, 1888185)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 811299 965590 322537 986788 646348 512299 574415 878476 709126 856456 674183 751119 158861 928050 691644 762819 345401 > 3²¹⁵ [i]