MinT - Best Known (133, 162, s)-Nets in Base 3

Best Known (133, 162, s)-Nets in Base 3

(133, 162, 700)-Net over F₃ — Constructive and digital

Digital (133, 162, 700)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (115, 144, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 36, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(133, 162, 3798)-Net over F₃ — Digital

Digital (133, 162, 3798)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶², 3798, F₃, 29) (dual of [3798, 3636, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶², 6579, F₃, 29) (dual of [6579, 6417, 30]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹⁴⁵, 6562, F₃, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. linear OA(3¹, 17, F₃, 1) (dual of [17, 16, 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]

(133, 162, 927504)-Net in Base 3 — Upper bound on s

There is no (133, 162, 927505)-net in base 3, because

1 times m-reduction [i] would yield (133, 161, 927505)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 65543 211343 617923 207642 430041 678730 283535 505369 179076 496409 279379 872546 626329 > 3¹⁶¹ [i]