MinT - Best Known (169, 169+35, s)-Nets in Base 3

Best Known (169, 169+35, s)-Nets in Base 3

(169, 169+35, 1480)-Net over F₃ — Constructive and digital

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]

(169, 169+35, 5635)-Net over F₃ — Digital

Digital (169, 204, 5635)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁴, 5635, F₃, 35) (dual of [5635, 5431, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁴, 6620, F₃, 35) (dual of [6620, 6416, 36]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3²⁰⁰, 6616, F₃, 35) (dual of [6616, 6416, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(27) [i] based on
      1. linear OA(3¹⁸⁵, 6561, F₃, 35) (dual of [6561, 6376, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      2. linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3¹⁵, 55, F₃, 6) (dual of [55, 40, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(169, 169+35, 1787723)-Net in Base 3 — Upper bound on s

There is no (169, 204, 1787724)-net in base 3, because

1 times m-reduction [i] would yield (169, 203, 1787724)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 171624 661979 861722 764476 081930 899853 788435 407370 308813 343658 235805 079736 409342 818175 387942 646937 > 3²⁰³ [i]