MinT - Best Known (187−29, 187, s)-Nets in Base 3

Best Known (187−29, 187, s)-Nets in Base 3

(187−29, 187, 1480)-Net over F₃ — Constructive and digital

Digital (158, 187, 1480)-net over F₃, using

t-expansion [i] based on digital (157, 187, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (157, 188, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 47, 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]

(187−29, 187, 10548)-Net over F₃ — Digital

Digital (158, 187, 10548)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁷, 10548, F₃, 29) (dual of [10548, 10361, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁷, 19726, F₃, 29) (dual of [19726, 19539, 30]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(3¹⁸¹, 19684, F₃, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹⁴⁵, 19684, F₃, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(3⁶, 42, F₃, 3) (dual of [42, 36, 4]-code or 42-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]

(187−29, 187, 6596651)-Net in Base 3 — Upper bound on s

There is no (158, 187, 6596652)-net in base 3, because

1 times m-reduction [i] would yield (158, 186, 6596652)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55533 374554 637928 427996 254443 598810 432944 904123 988613 896982 244474 968966 242171 510872 823113 > 3¹⁸⁶ [i]