MinT - Best Known (57, 57+19, s)-Nets in Base 3

Best Known (57, 57+19, s)-Nets in Base 3

(57, 57+19, 328)-Net over F₃ — Constructive and digital

Digital (57, 76, 328)-net over F₃, using

trace code for nets [i] based on digital (0, 19, 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]

(57, 57+19, 442)-Net over F₃ — Digital

Digital (57, 76, 442)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁶, 442, F₃, 19) (dual of [442, 366, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁶, 734, F₃, 19) (dual of [734, 658, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(3⁷³, 730, F₃, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(3⁶¹, 730, F₃, 15) (dual of [730, 669, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    3. linear OA(3³, 4, F₃, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
      - dual of repetition code with length 4 [i]
      - oval in PG(2,Â 3) [i]

(57, 57+19, 19612)-Net in Base 3 — Upper bound on s

There is no (57, 76, 19613)-net in base 3, because

1 times m-reduction [i] would yield (57, 75, 19613)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 608270 091269 379731 216067 049766 821403 > 3⁷⁵ [i]