MinT - Best Known (75, 93, s)-Nets in Base 3

Best Known (75, 93, s)-Nets in Base 3

(75, 93, 640)-Net over F₃ — Constructive and digital

Digital (75, 93, 640)-net over F₃, using

3¹ times duplication [i] based on digital (74, 92, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 23, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(75, 93, 1869)-Net over F₃ — Digital

Digital (75, 93, 1869)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹³, 1869, F₃, 18) (dual of [1869, 1776, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹³, 2224, F₃, 18) (dual of [2224, 2131, 19]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [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(3⁵⁷, 2188, F₃, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’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, 36, F₃, 4) (dual of [36, 28, 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]

(75, 93, 176584)-Net in Base 3 — Upper bound on s

There is no (75, 93, 176585)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 235 656461 205775 652844 742258 663973 967498 899955 > 3⁹³ [i]