MinT - Best Known (93, 93+30, s)-Nets in Base 3

Best Known (93, 93+30, s)-Nets in Base 3

(93, 93+30, 328)-Net over F₃ — Constructive and digital

Digital (93, 123, 328)-net over F₃, using

1 times m-reduction [i] based on digital (93, 124, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 31, 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]

(93, 93+30, 653)-Net over F₃ — Digital

Digital (93, 123, 653)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²³, 653, F₃, 30) (dual of [653, 530, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²³, 749, F₃, 30) (dual of [749, 626, 31]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(25) [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(3¹⁰³, 729, F₃, 26) (dual of [729, 626, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
      - doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]

(93, 93+30, 26235)-Net in Base 3 — Upper bound on s

There is no (93, 123, 26236)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48530 323964 057092 752142 806099 458306 103993 825119 420827 685841 > 3¹²³ [i]