MinT - Best Known (131, 131+22, s)-Nets in Base 3

Best Known (131, 131+22, s)-Nets in Base 3

(131, 131+22, 5375)-Net over F₃ — Constructive and digital

Digital (131, 153, 5375)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (119, 141, 5368)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁴¹, 5368, F₃, 22, 22) (dual of [(5368, 22), 117955, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3¹⁴¹, 59048, F₃, 22) (dual of [59048, 58907, 23]-code), using
      - discarding factors / shortening the dual code 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+22, 24532)-Net over F₃ — Digital

Digital (131, 153, 24532)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵³, 24532, F₃, 2, 22) (dual of [(24532, 2), 48911, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵³, 29550, F₃, 2, 22) (dual of [(29550, 2), 58947, 23]-NRT-code), using
  - 3¹ times duplication [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
        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]
        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]
        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+22, large)-Net in Base 3 — Upper bound on s

There is no (131, 153, large)-net in base 3, because

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