MinT - Best Known (150−12, 150, s)-Nets in Base 3

Best Known (150−12, 150, s)-Nets in Base 3

(150−12, 150, 1594334)-Net over F₃ — Constructive and digital

Digital (138, 150, 1594334)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 9)-net over F₃, using
  - 1 times m-reduction [i] based on digital (2, 9, 9)-net over F₃, using
    - a shift-net [i]
2. digital (130, 142, 1594325)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁴², 1594325, F₃, 14, 12) (dual of [(1594325, 14), 22320408, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(3¹⁴², 4782976, F₃, 2, 12) (dual of [(4782976, 2), 9565810, 13]-NRT-code), using
      - trace code [i] based on linear OOA(9⁷¹, 2391488, F₉, 2, 12) (dual of [(2391488, 2), 4782905, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(9⁷¹, 4782976, F₉, 12) (dual of [4782976, 4782905, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(9⁷¹, 4782969, F₉, 12) (dual of [4782969, 4782898, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(9⁶⁴, 4782969, F₉, 11) (dual of [4782969, 4782905, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(9⁰, 7, F₉, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(150−12, 150, large)-Net over F₃ — Digital

Digital (138, 150, large)-net over F₃, using

3¹¹ times duplication [i] based on digital (127, 139, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁹, large, F₃, 12) (dual of [large, large−139, 13]-code), using
  - 19 times code embedding in larger space [i] based on linear OA(3¹²⁰, large, F₃, 12) (dual of [large, large−120, 13]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(150−12, 150, large)-Net in Base 3 — Upper bound on s

There is no (138, 150, large)-net in base 3, because

10 times m-reduction [i] would yield (138, 140, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (150−12, 150, s)-Nets in Base 3

(150−12, 150, 1594334)-Net over F3 — Constructive and digital

(150−12, 150, large)-Net over F3 — Digital

(150−12, 150, large)-Net in Base 3 — Upper bound on s

(150−12, 150, 1594334)-Net over F₃ — Constructive and digital

(150−12, 150, large)-Net over F₃ — Digital