MinT - Best Known (36, 36+13, s)-Nets in Base 3

Best Known (36, 36+13, s)-Nets in Base 3

(36, 36+13, 156)-Net over F₃ — Constructive and digital

Digital (36, 49, 156)-net over F₃, using

3¹ times duplication [i] based on digital (35, 48, 156)-net over F₃, using
- trace code for nets [i] based on digital (3, 16, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]

(36, 36+13, 365)-Net over F₃ — Digital

Digital (36, 49, 365)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁴⁹, 365, F₃, 2, 13) (dual of [(365, 2), 681, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁴⁹, 730, F₃, 13) (dual of [730, 681, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(36, 36+13, 9815)-Net in Base 3 — Upper bound on s

There is no (36, 49, 9816)-net in base 3, because

1 times m-reduction [i] would yield (36, 48, 9816)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 79783 679280 370230 338577 > 3⁴⁸ [i]