MinT - Best Known (61−18, 61, s)-Nets in Base 3

Best Known (61−18, 61, s)-Nets in Base 3

(61−18, 61, 144)-Net over F₃ — Constructive and digital

Digital (43, 61, 144)-net over F₃, using

3¹ times duplication [i] based on digital (42, 60, 144)-net over F₃, using
- trace code for nets [i] based on digital (2, 20, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]

(61−18, 61, 197)-Net over F₃ — Digital

Digital (43, 61, 197)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶¹, 197, F₃, 18) (dual of [197, 136, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶¹, 242, F₃, 18) (dual of [242, 181, 19]-code), using
  - the primitive BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â {−4,−3,…,13}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(61−18, 61, 3544)-Net in Base 3 — Upper bound on s

There is no (43, 61, 3545)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 127468 955662 370344 846643 444499 > 3⁶¹ [i]