MinT - Best Known (83−11, 83, s)-Nets in Base 3

Best Known (83−11, 83, s)-Nets in Base 3

(83−11, 83, 35435)-Net over F₃ — Constructive and digital

Digital (72, 83, 35435)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (67, 78, 35431)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁷⁸, 35431, F₃, 11, 11) (dual of [(35431, 11), 389663, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(3⁷⁸, 177156, F₃, 11) (dual of [177156, 177078, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷⁸, 177158, F₃, 11) (dual of [177158, 177080, 12]-code), using
        constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
        linear OA(3⁷⁸, 177147, F₃, 11) (dual of [177147, 177069, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(3⁶⁷, 177147, F₃, 10) (dual of [177147, 177080, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(3⁰, 11, F₃, 0) (dual of [11, 11, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(83−11, 83, 88587)-Net over F₃ — Digital

Digital (72, 83, 88587)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁸³, 88587, F₃, 2, 11) (dual of [(88587, 2), 177091, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁸³, 177174, F₃, 11) (dual of [177174, 177091, 12]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3⁸², 177173, F₃, 11) (dual of [177173, 177091, 12]-code), using
    - constructionÂ X applied to C_e(10) âŠ‚ C_e(7) [i] based on
      1. linear OA(3⁷⁸, 177147, F₃, 11) (dual of [177147, 177069, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      2. linear OA(3⁵⁶, 177147, F₃, 8) (dual of [177147, 177091, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
      3. linear OA(3⁴, 26, F₃, 2) (dual of [26, 22, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(83−11, 83, large)-Net in Base 3 — Upper bound on s

There is no (72, 83, large)-net in base 3, because

9 times m-reduction [i] would yield (72, 74, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (83−11, 83, s)-Nets in Base 3

(83−11, 83, 35435)-Net over F3 — Constructive and digital

(83−11, 83, 88587)-Net over F3 — Digital

(83−11, 83, large)-Net in Base 3 — Upper bound on s

(83−11, 83, 35435)-Net over F₃ — Constructive and digital

(83−11, 83, 88587)-Net over F₃ — Digital