MinT - Best Known (101, 101+19, s)-Nets in Base 3

Best Known (101, 101+19, s)-Nets in Base 3

(101, 101+19, 2195)-Net over F₃ — Constructive and digital

Digital (101, 120, 2195)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (90, 109, 2187)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁹, 2187, F₃, 19, 19) (dual of [(2187, 19), 41444, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [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]

(101, 101+19, 9865)-Net over F₃ — Digital

Digital (101, 120, 9865)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹²⁰, 9865, F₃, 2, 19) (dual of [(9865, 2), 19610, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹²⁰, 19730, F₃, 19) (dual of [19730, 19610, 20]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹²⁰, 19731, F₃, 19) (dual of [19731, 19611, 20]-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(3¹¹, 47, F₃, 5) (dual of [47, 36, 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]

(101, 101+19, 4220110)-Net in Base 3 — Upper bound on s

There is no (101, 120, 4220111)-net in base 3, because

1 times m-reduction [i] would yield (101, 119, 4220111)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 599 003925 541419 567300 751679 331478 749352 567007 170547 614607 > 3¹¹⁹ [i]