MinT - Best Known (186, 186+37, s)-Nets in Base 3

Best Known (186, 186+37, s)-Nets in Base 3

(186, 186+37, 1480)-Net over F₃ — Constructive and digital

Digital (186, 223, 1480)-net over F₃, using

t-expansion [i] based on digital (184, 223, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 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]

(186, 186+37, 8512)-Net over F₃ — Digital

Digital (186, 223, 8512)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²³, 8512, F₃, 2, 37) (dual of [(8512, 2), 16801, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²³, 9854, F₃, 2, 37) (dual of [(9854, 2), 19485, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²²³, 19708, F₃, 37) (dual of [19708, 19485, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
      1. 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]
      2. linear OA(3¹⁹⁹, 19684, F₃, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      3. linear OA(3⁶, 24, F₃, 3) (dual of [24, 18, 4]-code or 24-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]

(186, 186+37, 2894629)-Net in Base 3 — Upper bound on s

There is no (186, 223, 2894630)-net in base 3, because

1 times m-reduction [i] would yield (186, 222, 2894630)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8335 299729 577507 987970 618914 694137 470034 042372 646506 318878 708698 664002 839986 751062 718700 270476 510484 216397 > 3²²² [i]