MinT - Best Known (131, 131+21, s)-Nets in Base 3

Best Known (131, 131+21, s)-Nets in Base 3

(131, 131+21, 5912)-Net over F₃ — Constructive and digital

Digital (131, 152, 5912)-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 (119, 140, 5904)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁴⁰, 5904, F₃, 21, 21) (dual of [(5904, 21), 123844, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁴⁰, 59041, F₃, 21) (dual of [59041, 58901, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁴⁰, 59048, F₃, 21) (dual of [59048, 58908, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁴¹, 59049, F₃, 22) (dual of [59049, 58908, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(131, 131+21, 29550)-Net over F₃ — Digital

Digital (131, 152, 29550)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵², 29550, F₃, 2, 21) (dual of [(29550, 2), 58948, 22]-NRT-code), using
- strength reduction [i] based on linear OOA(3¹⁵², 29550, F₃, 2, 22) (dual of [(29550, 2), 58948, 23]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁵², 59100, F₃, 22) (dual of [59100, 58948, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
      1. linear OA(3¹⁴¹, 59049, F₃, 22) (dual of [59049, 58908, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(3¹⁰¹, 59049, F₃, 16) (dual of [59049, 58948, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(3¹¹, 51, F₃, 5) (dual of [51, 40, 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]

(131, 131+21, large)-Net in Base 3 — Upper bound on s

There is no (131, 152, large)-net in base 3, because

19 times m-reduction [i] would yield (131, 133, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]