MinT - Best Known (198−35, 198, s)-Nets in Base 3

Best Known (198−35, 198, s)-Nets in Base 3

(198−35, 198, 896)-Net over F₃ — Constructive and digital

Digital (163, 198, 896)-net over F₃, using

2 times m-reduction [i] based on digital (163, 200, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 50, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(198−35, 198, 4609)-Net over F₃ — Digital

Digital (163, 198, 4609)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁸, 4609, F₃, 35) (dual of [4609, 4411, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁸, 6606, F₃, 35) (dual of [6606, 6408, 36]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3¹⁹⁶, 6604, F₃, 35) (dual of [6604, 6408, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(28) [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(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(3¹¹, 43, F₃, 5) (dual of [43, 32, 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]

(198−35, 198, 1213116)-Net in Base 3 — Upper bound on s

There is no (163, 198, 1213117)-net in base 3, because

1 times m-reduction [i] would yield (163, 197, 1213117)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9837 563242 629154 275727 970319 268593 063334 499850 712781 980876 145043 600832 188679 233485 601410 223675 > 3¹⁹⁷ [i]