MinT - Best Known (90, 90+30, s)-Nets in Base 3

Best Known (90, 90+30, s)-Nets in Base 3

(90, 90+30, 328)-Net over F₃ — Constructive and digital

Digital (90, 120, 328)-net over F₃, using

trace code for nets [i] based on digital (0, 30, 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]

(90, 90+30, 578)-Net over F₃ — Digital

Digital (90, 120, 578)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁰, 578, F₃, 30) (dual of [578, 458, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁰, 740, F₃, 30) (dual of [740, 620, 31]-code), using
  - 1 times truncation [i] based on linear OA(3¹²¹, 741, F₃, 31) (dual of [741, 620, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
      1. linear OA(3¹¹⁸, 729, F₃, 31) (dual of [729, 611, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3³, 12, F₃, 2) (dual of [12, 9, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]

(90, 90+30, 21057)-Net in Base 3 — Upper bound on s

There is no (90, 120, 21058)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1797 587211 423437 993691 089376 213409 543283 235030 587568 694681 > 3¹²⁰ [i]