MinT - Best Known (123, 123+31, s)-Nets in Base 3

Best Known (123, 123+31, s)-Nets in Base 3

(123, 123+31, 688)-Net over F₃ — Constructive and digital

Digital (123, 154, 688)-net over F₃, using

3² times duplication [i] based on digital (121, 152, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 38, 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]

(123, 123+31, 1893)-Net over F₃ — Digital

Digital (123, 154, 1893)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁴, 1893, F₃, 31) (dual of [1893, 1739, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁴, 2229, F₃, 31) (dual of [2229, 2075, 32]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3¹⁵¹, 2226, F₃, 31) (dual of [2226, 2075, 32]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      2. linear OA(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(123, 123+31, 236236)-Net in Base 3 — Upper bound on s

There is no (123, 154, 236237)-net in base 3, because

1 times m-reduction [i] would yield (123, 153, 236237)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 990079 228886 273066 689508 584362 082633 479326 482306 857228 031472 231976 484251 > 3¹⁵³ [i]