MinT - Best Known (185−29, 185, s)-Nets in Base 3

Best Known (185−29, 185, s)-Nets in Base 3

(185−29, 185, 1480)-Net over F₃ — Constructive and digital

Digital (156, 185, 1480)-net over F₃, using

3¹ times duplication [i] based on digital (155, 184, 1480)-net over F₃, using
- t-expansion [i] based on digital (154, 184, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 46, 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]

(185−29, 185, 9866)-Net over F₃ — Digital

Digital (156, 185, 9866)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁵, 9866, F₃, 2, 29) (dual of [(9866, 2), 19547, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁸³, 9865, F₃, 2, 29) (dual of [(9865, 2), 19547, 30]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁸³, 19730, F₃, 29) (dual of [19730, 19547, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(22) [i] based on
      1. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(3¹³⁶, 19683, F₃, 23) (dual of [19683, 19547, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(185−29, 185, 5638494)-Net in Base 3 — Upper bound on s

There is no (156, 185, 5638495)-net in base 3, because

1 times m-reduction [i] would yield (156, 184, 5638495)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6170 366204 798697 006519 132545 841160 417575 807442 994890 043487 778617 266119 145954 780251 045797 > 3¹⁸⁴ [i]