MinT - Best Known (218, 218+29, s)-Nets in Base 3

Best Known (218, 218+29, s)-Nets in Base 3

(218, 218+29, 37972)-Net over F₃ — Constructive and digital

Digital (218, 247, 37972)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (200, 229, 37960)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²²⁹, 37960, F₃, 29, 29) (dual of [(37960, 29), 1100611, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(3²²⁹, 531441, F₃, 29) (dual of [531441, 531212, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(218, 218+29, 177173)-Net over F₃ — Digital

Digital (218, 247, 177173)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁷, 177173, F₃, 3, 29) (dual of [(177173, 3), 531272, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3²⁴⁴, 177172, F₃, 3, 29) (dual of [(177172, 3), 531272, 30]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²⁴⁴, 531516, F₃, 29) (dual of [531516, 531272, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(21) [i] based on
      1. linear OA(3²²⁹, 531441, F₃, 29) (dual of [531441, 531212, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(3¹⁵, 75, F₃, 6) (dual of [75, 60, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(218, 218+29, large)-Net in Base 3 — Upper bound on s

There is no (218, 247, large)-net in base 3, because

27 times m-reduction [i] would yield (218, 220, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]