MinT - Best Known (166−21, 166, s)-Nets in Base 3

Best Known (166−21, 166, s)-Nets in Base 3

(166−21, 166, 17722)-Net over F₃ — Constructive and digital

Digital (145, 166, 17722)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (133, 154, 17714)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵⁴, 17714, F₃, 21, 21) (dual of [(17714, 21), 371840, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁵⁴, 177141, F₃, 21) (dual of [177141, 176987, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵⁴, 177146, F₃, 21) (dual of [177146, 176992, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁵⁵, 177147, F₃, 22) (dual of [177147, 176992, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(166−21, 166, 83969)-Net over F₃ — Digital

Digital (145, 166, 83969)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁶, 83969, F₃, 2, 21) (dual of [(83969, 2), 167772, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁶⁶, 88601, F₃, 2, 21) (dual of [(88601, 2), 177036, 22]-NRT-code), using
  - strength reduction [i] based on linear OOA(3¹⁶⁶, 88601, F₃, 2, 22) (dual of [(88601, 2), 177036, 23]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁶⁶, 177202, F₃, 22) (dual of [177202, 177036, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
        linear OA(3¹⁵⁵, 177147, F₃, 22) (dual of [177147, 176992, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(3¹¹¹, 177147, F₃, 16) (dual of [177147, 177036, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(3¹¹, 55, F₃, 5) (dual of [55, 44, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(166−21, 166, large)-Net in Base 3 — Upper bound on s

There is no (145, 166, large)-net in base 3, because

19 times m-reduction [i] would yield (145, 147, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]