MinT - Best Known (75, 99, s)-Nets in Base 3

Best Known (75, 99, s)-Nets in Base 3

(75, 99, 328)-Net over F₃ — Constructive and digital

Digital (75, 99, 328)-net over F₃, using

1 times m-reduction [i] based on digital (75, 100, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 25, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(75, 99, 585)-Net over F₃ — Digital

Digital (75, 99, 585)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁹, 585, F₃, 24) (dual of [585, 486, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁹, 734, F₃, 24) (dual of [734, 635, 25]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(3⁹⁷, 730, F₃, 25) (dual of [730, 633, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(3⁸⁵, 730, F₃, 21) (dual of [730, 645, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    3. linear OA(3², 4, F₃, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
      - extended Reedâ€“Solomon code RS_e(2,3) [i]
      - Hamming code H(2,3) [i]
      - Simplex code S(2,3) [i]
      - the Tetracode [i]

(75, 99, 22822)-Net in Base 3 — Upper bound on s

There is no (75, 99, 22823)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 171838 513420 324283 200684 342970 949001 165757 784297 > 3⁹⁹ [i]