MinT - Best Known (164−23, 164, s)-Nets in Base 3

Best Known (164−23, 164, s)-Nets in Base 3

(164−23, 164, 5376)-Net over F₃ — Constructive and digital

Digital (141, 164, 5376)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 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 (128, 151, 5368)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵¹, 5368, F₃, 23, 23) (dual of [(5368, 23), 123313, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(164−23, 164, 29551)-Net over F₃ — Digital

Digital (141, 164, 29551)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁴, 29551, F₃, 2, 23) (dual of [(29551, 2), 58938, 24]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁶², 29550, F₃, 2, 23) (dual of [(29550, 2), 58938, 24]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁶², 59100, F₃, 23) (dual of [59100, 58938, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(16) [i] based on
      1. linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(3¹¹¹, 59049, F₃, 17) (dual of [59049, 58938, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(164−23, 164, large)-Net in Base 3 — Upper bound on s

There is no (141, 164, large)-net in base 3, because

21 times m-reduction [i] would yield (141, 143, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]