MinT - Best Known (119−17, 119, s)-Nets in Base 3

Best Known (119−17, 119, s)-Nets in Base 3

(119−17, 119, 7386)-Net over F₃ — Constructive and digital

Digital (102, 119, 7386)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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 (94, 111, 7382)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹¹¹, 7382, F₃, 17, 17) (dual of [(7382, 17), 125383, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹¹¹, 59057, F₃, 17) (dual of [59057, 58946, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹¹¹, 59059, F₃, 17) (dual of [59059, 58948, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
        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]
        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(3⁰, 10, F₃, 0) (dual of [10, 10, 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]

(119−17, 119, 29350)-Net over F₃ — Digital

Digital (102, 119, 29350)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁹, 29350, F₃, 2, 17) (dual of [(29350, 2), 58581, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹¹⁹, 29543, F₃, 2, 17) (dual of [(29543, 2), 58967, 18]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹¹⁸, 29543, F₃, 2, 17) (dual of [(29543, 2), 58968, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹¹⁸, 59086, F₃, 17) (dual of [59086, 58968, 18]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹¹⁷, 59085, F₃, 17) (dual of [59085, 58968, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
        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]
        linear OA(3⁸¹, 59049, F₃, 13) (dual of [59049, 58968, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁶, 36, F₃, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(119−17, 119, large)-Net in Base 3 — Upper bound on s

There is no (102, 119, large)-net in base 3, because

15 times m-reduction [i] would yield (102, 104, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (119−17, 119, s)-Nets in Base 3

(119−17, 119, 7386)-Net over F3 — Constructive and digital

(119−17, 119, 29350)-Net over F3 — Digital

(119−17, 119, large)-Net in Base 3 — Upper bound on s

(119−17, 119, 7386)-Net over F₃ — Constructive and digital

(119−17, 119, 29350)-Net over F₃ — Digital