MinT - Best Known (75, 75+19, s)-Nets in Base 3

Best Known (75, 75+19, s)-Nets in Base 3

(75, 75+19, 600)-Net over F₃ — Constructive and digital

Digital (75, 94, 600)-net over F₃, using

3² times duplication [i] based on digital (73, 92, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 23, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(75, 75+19, 1447)-Net over F₃ — Digital

Digital (75, 94, 1447)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁴, 1447, F₃, 19) (dual of [1447, 1353, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁴, 2198, F₃, 19) (dual of [2198, 2104, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([1,9]) [i] based on
    1. linear OA(3⁸⁵, 2188, F₃, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(3⁸⁴, 2188, F₃, 9) (dual of [2188, 2104, 10]-code), using the narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(3⁹, 10, F₃, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
      - dual of repetition code with length 10 [i]

(75, 75+19, 176584)-Net in Base 3 — Upper bound on s

There is no (75, 94, 176585)-net in base 3, because

1 times m-reduction [i] would yield (75, 93, 176585)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 235 656461 205775 652844 742258 663973 967498 899955 > 3⁹³ [i]