MinT - Best Known (71, 94, s)-Nets in Base 3

Best Known (71, 94, s)-Nets in Base 3

(71, 94, 328)-Net over F₃ — Constructive and digital

Digital (71, 94, 328)-net over F₃, using

3² times duplication [i] based on digital (69, 92, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 23, 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]

(71, 94, 545)-Net over F₃ — Digital

Digital (71, 94, 545)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁴, 545, F₃, 23) (dual of [545, 451, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁴, 742, F₃, 23) (dual of [742, 648, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
    1. linear OA(3⁹¹, 729, F₃, 23) (dual of [729, 638, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(3⁷⁹, 729, F₃, 20) (dual of [729, 650, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(71, 94, 26525)-Net in Base 3 — Upper bound on s

There is no (71, 94, 26526)-net in base 3, because

1 times m-reduction [i] would yield (71, 93, 26526)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 235 751235 627127 823998 622047 041912 360920 309297 > 3⁹³ [i]