MinT - Best Known (142−17, 142, s)-Nets in Base 3

Best Known (142−17, 142, s)-Nets in Base 3

(142−17, 142, 66438)-Net over F₃ — Constructive and digital

Digital (125, 142, 66438)-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 (116, 133, 66431)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³³, 66431, F₃, 17, 17) (dual of [(66431, 17), 1129194, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹³³, 531449, F₃, 17) (dual of [531449, 531316, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³³, 531453, F₃, 17) (dual of [531453, 531320, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
        linear OA(3¹³³, 531441, F₃, 17) (dual of [531441, 531308, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(3¹²¹, 531441, F₃, 16) (dual of [531441, 531320, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(3⁰, 12, F₃, 0) (dual of [12, 12, 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]

(142−17, 142, 178488)-Net over F₃ — Digital

Digital (125, 142, 178488)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴², 178488, F₃, 2, 17) (dual of [(178488, 2), 356834, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁴², 265743, F₃, 2, 17) (dual of [(265743, 2), 531344, 18]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁴⁰, 265742, F₃, 2, 17) (dual of [(265742, 2), 531344, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁴⁰, 531484, F₃, 17) (dual of [531484, 531344, 18]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(3¹³⁹, 531483, F₃, 17) (dual of [531483, 531344, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
        linear OA(3¹³³, 531441, F₃, 17) (dual of [531441, 531308, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(3⁹⁷, 531441, F₃, 13) (dual of [531441, 531344, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁶, 42, F₃, 3) (dual of [42, 36, 4]-code or 42-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]

(142−17, 142, large)-Net in Base 3 — Upper bound on s

There is no (125, 142, large)-net in base 3, because

15 times m-reduction [i] would yield (125, 127, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (142−17, 142, s)-Nets in Base 3

(142−17, 142, 66438)-Net over F3 — Constructive and digital

(142−17, 142, 178488)-Net over F3 — Digital

(142−17, 142, large)-Net in Base 3 — Upper bound on s

(142−17, 142, 66438)-Net over F₃ — Constructive and digital

(142−17, 142, 178488)-Net over F₃ — Digital