MinT - Best Known (125−31, 125, s)-Nets in Base 3

Best Known (125−31, 125, s)-Nets in Base 3

(125−31, 125, 328)-Net over F₃ — Constructive and digital

Digital (94, 125, 328)-net over F₃, using

3¹ times duplication [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]

(125−31, 125, 615)-Net over F₃ — Digital

Digital (94, 125, 615)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁵, 615, F₃, 31) (dual of [615, 490, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁵, 740, F₃, 31) (dual of [740, 615, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹²¹, 730, F₃, 31) (dual of [730, 609, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹⁰⁹, 730, F₃, 27) (dual of [730, 621, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (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]

(125−31, 125, 28230)-Net in Base 3 — Upper bound on s

There is no (94, 125, 28231)-net in base 3, because

1 times m-reduction [i] would yield (94, 124, 28231)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 145609 130899 023563 947784 698585 414040 875218 432642 074367 404291 > 3¹²⁴ [i]