MinT - Best Known (40, 40+13, s)-Nets in Base 3

Best Known (40, 40+13, s)-Nets in Base 3

(40, 40+13, 328)-Net over F₃ — Constructive and digital

Digital (40, 53, 328)-net over F₃, using

3¹ times duplication [i] based on digital (39, 52, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 13, 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]

(40, 40+13, 433)-Net over F₃ — Digital

Digital (40, 53, 433)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵³, 433, F₃, 13) (dual of [433, 380, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵³, 740, F₃, 13) (dual of [740, 687, 14]-code), using
  - constructionÂ X applied to C([0,6]) âŠ‚ C([0,4]) [i] based on
    1. linear OA(3⁴⁹, 730, F₃, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    2. linear OA(3³⁷, 730, F₃, 9) (dual of [730, 693, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    3. 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]

(40, 40+13, 20423)-Net in Base 3 — Upper bound on s

There is no (40, 53, 20424)-net in base 3, because

1 times m-reduction [i] would yield (40, 52, 20424)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 462399 826726 763750 702769 > 3⁵² [i]