MinT - Best Known (178−33, 178, s)-Nets in Base 3

Best Known (178−33, 178, s)-Nets in Base 3

(178−33, 178, 696)-Net over F₃ — Constructive and digital

Digital (145, 178, 696)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (127, 160, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 40, 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]

(178−33, 178, 3289)-Net over F₃ — Digital

Digital (145, 178, 3289)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁸, 3289, F₃, 2, 33) (dual of [(3289, 2), 6400, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁷⁸, 6578, F₃, 33) (dual of [6578, 6400, 34]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁷⁸, 6579, F₃, 33) (dual of [6579, 6401, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,15]) [i] based on
      1. linear OA(3¹⁷⁷, 6562, F₃, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. 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]
      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]

(178−33, 178, 645137)-Net in Base 3 — Upper bound on s

There is no (145, 178, 645138)-net in base 3, because

1 times m-reduction [i] would yield (145, 177, 645138)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 821429 859642 862056 014683 618008 680030 983482 673452 796980 184080 044496 610127 166526 957217 > 3¹⁷⁷ [i]