MinT - Best Known (88, 113, s)-Nets in Base 3

Best Known (88, 113, s)-Nets in Base 3

(88, 113, 464)-Net over F₃ — Constructive and digital

Digital (88, 113, 464)-net over F₃, using

3¹ times duplication [i] based on digital (87, 112, 464)-net over F₃, using
- t-expansion [i] based on digital (86, 112, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 28, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(88, 113, 1094)-Net over F₃ — Digital

Digital (88, 113, 1094)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹³, 1094, F₃, 2, 25) (dual of [(1094, 2), 2075, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(88, 113, 75057)-Net in Base 3 — Upper bound on s

There is no (88, 113, 75058)-net in base 3, because

1 times m-reduction [i] would yield (88, 112, 75058)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 273895 698473 926130 454345 754348 267236 777245 538421 833337 > 3¹¹² [i]