MinT - Best Known (180, 180+35, s)-Nets in Base 3

Best Known (180, 180+35, s)-Nets in Base 3

(180, 180+35, 1480)-Net over F₃ — Constructive and digital

Digital (180, 215, 1480)-net over F₃, using

t-expansion [i] based on digital (178, 215, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (178, 216, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 54, 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]

(180, 180+35, 9556)-Net over F₃ — Digital

Digital (180, 215, 9556)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁵, 9556, F₃, 2, 35) (dual of [(9556, 2), 18897, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁵, 9858, F₃, 2, 35) (dual of [(9858, 2), 19501, 36]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3²¹⁴, 9858, F₃, 2, 35) (dual of [(9858, 2), 19502, 36]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹⁴, 19716, F₃, 35) (dual of [19716, 19502, 36]-code), using
      - constructionÂ X applied to C_e(34) âŠ‚ C_e(30) [i] based on
        linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
        linear OA(3¹⁸¹, 19683, F₃, 31) (dual of [19683, 19502, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
        linear OA(3⁶, 33, F₃, 3) (dual of [33, 27, 4]-code or 33-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]

(180, 180+35, 3639383)-Net in Base 3 — Upper bound on s

There is no (180, 215, 3639384)-net in base 3, because

1 times m-reduction [i] would yield (180, 214, 3639384)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 270424 522163 000835 723498 313825 492790 127755 835665 505130 006316 722772 837918 745752 892783 755182 631978 315697 > 3²¹⁴ [i]