MinT - Best Known (118−18, 118, s)-Nets in Base 3

Best Known (118−18, 118, s)-Nets in Base 3

(118−18, 118, 2194)-Net over F₃ — Constructive and digital

Digital (100, 118, 2194)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (90, 108, 2187)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁸, 2187, F₃, 18, 18) (dual of [(2187, 18), 39258, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁰⁸, 19683, F₃, 18) (dual of [19683, 19575, 19]-code), using
      - 1 times truncation [i] based on linear OA(3¹⁰⁹, 19684, F₃, 19) (dual of [19684, 19575, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(118−18, 118, 10467)-Net over F₃ — Digital

Digital (100, 118, 10467)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁸, 10467, F₃, 18) (dual of [10467, 10349, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁸, 19729, F₃, 18) (dual of [19729, 19611, 19]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(3¹⁰⁹, 19684, F₃, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(3⁷³, 19684, F₃, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    3. linear OA(3⁹, 45, F₃, 4) (dual of [45, 36, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(118−18, 118, 3735168)-Net in Base 3 — Upper bound on s

There is no (100, 118, 3735169)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 199 667845 389276 234294 926439 180314 714450 181444 824325 976547 > 3¹¹⁸ [i]