MinT - Best Known (188−35, 188, s)-Nets in Base 3

Best Known (188−35, 188, s)-Nets in Base 3

(188−35, 188, 698)-Net over F₃ — Constructive and digital

Digital (153, 188, 698)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (133, 168, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 42, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(188−35, 188, 3295)-Net over F₃ — Digital

Digital (153, 188, 3295)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁸, 3295, F₃, 35) (dual of [3295, 3107, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁸, 6574, F₃, 35) (dual of [6574, 6386, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(31) [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(3¹⁶⁹, 6561, F₃, 32) (dual of [6561, 6392, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(188−35, 188, 635678)-Net in Base 3 — Upper bound on s

There is no (153, 188, 635679)-net in base 3, because

1 times m-reduction [i] would yield (153, 187, 635679)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 166600 668713 657761 578002 999658 795646 215379 369986 452507 686129 321685 934259 816459 494983 054879 > 3¹⁸⁷ [i]