MinT - Best Known (158−39, 158, s)-Nets in Base 3

Best Known (158−39, 158, s)-Nets in Base 3

(158−39, 158, 328)-Net over F₃ — Constructive and digital

Digital (119, 158, 328)-net over F₃, using

3² times duplication [i] based on digital (117, 156, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 39, 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]

(158−39, 158, 743)-Net over F₃ — Digital

Digital (119, 158, 743)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁸, 743, F₃, 39) (dual of [743, 585, 40]-code), using
- constructionÂ X applied to C([0,19]) âŠ‚ C([0,18]) [i] based on
  1. linear OA(3¹⁵⁷, 730, F₃, 39) (dual of [730, 573, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
  2. 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]
  3. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(158−39, 158, 34712)-Net in Base 3 — Upper bound on s

There is no (119, 158, 34713)-net in base 3, because

1 times m-reduction [i] would yield (119, 157, 34713)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 809 508812 727979 425130 457020 171818 975162 466619 592330 151943 945066 289572 766459 > 3¹⁵⁷ [i]