MinT - Best Known (213−33, 213, s)-Nets in Base 3

Best Known (213−33, 213, s)-Nets in Base 3

(213−33, 213, 1487)-Net over F₃ — Constructive and digital

Digital (180, 213, 1487)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (163, 196, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 49, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(213−33, 213, 11344)-Net over F₃ — Digital

Digital (180, 213, 11344)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹³, 11344, F₃, 33) (dual of [11344, 11131, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹³, 19736, F₃, 33) (dual of [19736, 19523, 34]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(25) [i] based on
    1. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹⁵⁴, 19683, F₃, 26) (dual of [19683, 19529, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
      - codes constructed by inverting constructionÂ Y₁ [i]

(213−33, 213, 7134514)-Net in Base 3 — Upper bound on s

There is no (180, 213, 7134515)-net in base 3, because

1 times m-reduction [i] would yield (180, 212, 7134515)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141158 397441 543155 361884 377232 822625 092387 205970 617075 460338 135318 851298 467029 446247 171304 541486 875361 > 3²¹² [i]