MinT - Best Known (233−47, 233, s)-Nets in Base 3

Best Known (233−47, 233, s)-Nets in Base 3

(233−47, 233, 688)-Net over F₃ — Constructive and digital

Digital (186, 233, 688)-net over F₃, using

t-expansion [i] based on digital (184, 233, 688)-net over F₃, using
- 3 times m-reduction [i] based on digital (184, 236, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 59, 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]

(233−47, 233, 2388)-Net over F₃ — Digital

Digital (186, 233, 2388)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³³, 2388, F₃, 47) (dual of [2388, 2155, 48]-code), using
- 179 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0, 1, 28 times 0, 1, 34 times 0, 1, 42 times 0) [i] based on linear OA(3²¹⁸, 2194, F₃, 47) (dual of [2194, 1976, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(45) [i] based on
    1. linear OA(3²¹⁸, 2187, F₃, 47) (dual of [2187, 1969, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(3²¹¹, 2187, F₃, 46) (dual of [2187, 1976, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(233−47, 233, 306263)-Net in Base 3 — Upper bound on s

There is no (186, 233, 306264)-net in base 3, because

1 times m-reduction [i] would yield (186, 232, 306264)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 492 193774 880878 042945 198147 222929 129038 965058 148590 036995 032452 486488 159855 209936 558523 540308 810053 764839 989025 > 3²³² [i]