MinT - Best Known (57, 57+17, s)-Nets in Base 3

Best Known (57, 57+17, s)-Nets in Base 3

(57, 57+17, 400)-Net over F₃ — Constructive and digital

Digital (57, 74, 400)-net over F₃, using

3² times duplication [i] based on digital (55, 72, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 18, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(57, 57+17, 661)-Net over F₃ — Digital

Digital (57, 74, 661)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁴, 661, F₃, 17) (dual of [661, 587, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁴, 743, F₃, 17) (dual of [743, 669, 18]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(3⁷³, 730, F₃, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(3⁶¹, 730, F₃, 15) (dual of [730, 669, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (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]

(57, 57+17, 42492)-Net in Base 3 — Upper bound on s

There is no (57, 74, 42493)-net in base 3, because

1 times m-reduction [i] would yield (57, 73, 42493)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 67588 329303 084948 051712 652426 231649 > 3⁷³ [i]