MinT - Best Known (188, 223, s)-Nets in Base 3

Best Known (188, 223, s)-Nets in Base 3

(188, 223, 1488)-Net over F₃ — Constructive and digital

Digital (188, 223, 1488)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (169, 204, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 51, 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]

(188, 223, 10635)-Net over F₃ — Digital

Digital (188, 223, 10635)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²³, 10635, F₃, 35) (dual of [10635, 10412, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²³, 19726, F₃, 35) (dual of [19726, 19503, 36]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [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(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]
    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]

(188, 223, 6103181)-Net in Base 3 — Upper bound on s

There is no (188, 223, 6103182)-net in base 3, because

1 times m-reduction [i] would yield (188, 222, 6103182)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8335 253808 820862 404388 956174 018105 250013 044678 632642 877923 335951 461647 889805 120523 621311 537912 162145 115901 > 3²²² [i]