MinT - Best Known (216−36, 216, s)-Nets in Base 3

Best Known (216−36, 216, s)-Nets in Base 3

(216−36, 216, 1480)-Net over F₃ — Constructive and digital

Digital (180, 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]

(216−36, 216, 8141)-Net over F₃ — Digital

Digital (180, 216, 8141)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁶, 8141, F₃, 2, 36) (dual of [(8141, 2), 16066, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁶, 9841, F₃, 2, 36) (dual of [(9841, 2), 19466, 37]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²¹⁶, 19682, F₃, 36) (dual of [19682, 19466, 37]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²¹⁶, 19683, F₃, 36) (dual of [19683, 19467, 37]-code), using
      - 1 times truncation [i] based on linear OA(3²¹⁷, 19684, F₃, 37) (dual of [19684, 19467, 38]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]

(216−36, 216, 2007018)-Net in Base 3 — Upper bound on s

There is no (180, 216, 2007019)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 433883 768659 576023 022834 751898 470849 801396 943903 020109 325865 230644 371770 508301 103684 657083 559122 370645 > 3²¹⁶ [i]