MinT - Best Known (164, 164+35, s)-Nets in Base 3

Best Known (164, 164+35, s)-Nets in Base 3

(164, 164+35, 896)-Net over F₃ — Constructive and digital

Digital (164, 199, 896)-net over F₃, using

t-expansion [i] based on digital (163, 199, 896)-net over F₃, using
- 1 times m-reduction [i] based on digital (163, 200, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 50, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(164, 164+35, 4766)-Net over F₃ — Digital

Digital (164, 199, 4766)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁹, 4766, F₃, 35) (dual of [4766, 4567, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁹, 6600, F₃, 35) (dual of [6600, 6401, 36]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
    1. linear OA(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    3. linear OA(3⁶, 38, F₃, 3) (dual of [38, 32, 4]-code or 38-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(164, 164+35, 1294103)-Net in Base 3 — Upper bound on s

There is no (164, 199, 1294104)-net in base 3, because

1 times m-reduction [i] would yield (164, 198, 1294104)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29512 880363 296021 899566 165755 092365 609991 287926 614543 516955 570982 734071 609960 711274 753442 366769 > 3¹⁹⁸ [i]