MinT - Best Known (106−18, 106, s)-Nets in Base 3

Best Known (106−18, 106, s)-Nets in Base 3

(106−18, 106, 736)-Net over F₃ — Constructive and digital

Digital (88, 106, 736)-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 (78, 96, 729)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹⁶, 729, F₃, 18, 18) (dual of [(729, 18), 13026, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3⁹⁶, 6561, F₃, 18) (dual of [6561, 6465, 19]-code), using
      - 1 times truncation [i] based on linear OA(3⁹⁷, 6562, F₃, 19) (dual of [6562, 6465, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(106−18, 106, 4584)-Net over F₃ — Digital

Digital (88, 106, 4584)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁶, 4584, F₃, 18) (dual of [4584, 4478, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, 6603, F₃, 18) (dual of [6603, 6497, 19]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3¹⁰⁵, 6602, F₃, 18) (dual of [6602, 6497, 19]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(3⁹⁷, 6562, F₃, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(3⁶⁵, 6562, F₃, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
      3. linear OA(3⁸, 40, F₃, 4) (dual of [40, 32, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁸, 41, F₃, 4) (dual of [41, 33, 5]-code), using
        the narrow-sense BCH-code C(I) with length 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(106−18, 106, 863267)-Net in Base 3 — Upper bound on s

There is no (88, 106, 863268)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 375 712977 962204 111280 704700 740822 187959 939545 943881 > 3¹⁰⁶ [i]