MinT - Best Known (233−46, 233, s)-Nets in Base 3

Best Known (233−46, 233, s)-Nets in Base 3

Digital (187, 233, 688)-net over F₃, using

7 times m-reduction [i] based on digital (187, 240, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 60, 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]

Digital (187, 233, 2628)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³³, 2628, F₃, 46) (dual of [2628, 2395, 47]-code), using
- 419 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0, 1, 28 times 0, 1, 35 times 0, 1, 41 times 0, 1, 46 times 0, 1, 51 times 0, 1, 55 times 0, 1, 59 times 0) [i] based on 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]

There is no (187, 233, 321249)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1476 576914 216468 127570 481506 748468 203194 712957 375765 242977 892595 902406 626981 290229 568225 443018 661060 663919 274955 > 3²³³ [i]