MinT - Best Known (238−19, 238, s)-Nets in Base 3

Best Known (238−19, 238, s)-Nets in Base 3

(238−19, 238, 1062892)-Net over F₃ — Constructive and digital

Digital (219, 238, 1062892)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (207, 226, 1062882)-net over F₃, using
  - trace code for nets [i] based on digital (94, 113, 531441)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9¹¹³, 531441, F₉, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
      - OOA 9-folding and stacking with additional row [i] based on linear OA(9¹¹³, 4782970, F₉, 19) (dual of [4782970, 4782857, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 9¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(238−19, 238, large)-Net over F₃ — Digital

Digital (219, 238, large)-net over F₃, using

3¹⁰ times duplication [i] based on digital (209, 228, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, large, F₃, 19) (dual of [large, large−228, 20]-code), using
  - 47 times code embedding in larger space [i] based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(238−19, 238, large)-Net in Base 3 — Upper bound on s

There is no (219, 238, large)-net in base 3, because

17 times m-reduction [i] would yield (219, 221, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]