MinT - Best Known (149−37, 149, s)-Nets in Base 3

Best Known (149−37, 149, s)-Nets in Base 3

(149−37, 149, 328)-Net over F₃ — Constructive and digital

Digital (112, 149, 328)-net over F₃, using

3¹ times duplication [i] based on digital (111, 148, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 37, 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]

(149−37, 149, 694)-Net over F₃ — Digital

Digital (112, 149, 694)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁹, 694, F₃, 37) (dual of [694, 545, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁹, 740, F₃, 37) (dual of [740, 591, 38]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(3¹⁴⁵, 730, F₃, 37) (dual of [730, 585, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(3¹³³, 730, F₃, 33) (dual of [730, 597, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (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]

(149−37, 149, 31612)-Net in Base 3 — Upper bound on s

There is no (112, 149, 31613)-net in base 3, because

1 times m-reduction [i] would yield (112, 148, 31613)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41125 096297 703818 411447 379122 851326 509433 082807 151570 758493 770151 909441 > 3¹⁴⁸ [i]