MinT - Best Known (150−16, 150, s)-Nets in Base 3

Best Known (150−16, 150, s)-Nets in Base 3

(150−16, 150, 597878)-Net over F₃ — Constructive and digital

Digital (134, 150, 597878)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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 (125, 141, 597871)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁴¹, 597871, F₃, 16, 16) (dual of [(597871, 16), 9565795, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(3¹⁴¹, 4782968, F₃, 16) (dual of [4782968, 4782827, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁴¹, 4782969, F₃, 16) (dual of [4782969, 4782828, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(150−16, 150, 1594340)-Net over F₃ — Digital

Digital (134, 150, 1594340)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁰, 1594340, F₃, 3, 16) (dual of [(1594340, 3), 4782870, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁵⁰, 4783020, F₃, 16) (dual of [4783020, 4782870, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
    1. linear OA(3¹⁴¹, 4782969, F₃, 16) (dual of [4782969, 4782828, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(3⁹⁹, 4782969, F₃, 11) (dual of [4782969, 4782870, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁹, 51, F₃, 4) (dual of [51, 42, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(150−16, 150, large)-Net in Base 3 — Upper bound on s

There is no (134, 150, large)-net in base 3, because

14 times m-reduction [i] would yield (134, 136, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]