MinT - Best Known (216−28, 216, s)-Nets in Base 3

Best Known (216−28, 216, s)-Nets in Base 3

(216−28, 216, 12663)-Net over F₃ — Constructive and digital

Digital (188, 216, 12663)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (171, 199, 12653)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁹, 12653, F₃, 28, 28) (dual of [(12653, 28), 354085, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(3¹⁹⁹, 177142, F₃, 28) (dual of [177142, 176943, 29]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁹, 177147, F₃, 28) (dual of [177147, 176948, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(216−28, 216, 61743)-Net over F₃ — Digital

Digital (188, 216, 61743)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁶, 61743, F₃, 2, 28) (dual of [(61743, 2), 123270, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁶, 88609, F₃, 2, 28) (dual of [(88609, 2), 177002, 29]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3²¹⁵, 88609, F₃, 2, 28) (dual of [(88609, 2), 177003, 29]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹⁵, 177218, F₃, 28) (dual of [177218, 177003, 29]-code), using
      - constructionÂ X applied to C_e(27) âŠ‚ C_e(19) [i] based on
        linear OA(3¹⁹⁹, 177147, F₃, 28) (dual of [177147, 176948, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(3¹⁴⁴, 177147, F₃, 20) (dual of [177147, 177003, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(3¹⁶, 71, F₃, 7) (dual of [71, 55, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(216−28, 216, large)-Net in Base 3 — Upper bound on s

There is no (188, 216, large)-net in base 3, because

26 times m-reduction [i] would yield (188, 190, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]