MinT - Best Known (226, 226+15, s)-Nets in Base 3

Best Known (226, 226+15, s)-Nets in Base 3

(226, 226+15, 4793510)-Net over F₃ — Constructive and digital

Digital (226, 241, 4793510)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (6, 13, 26)-net over F₃, using
  - linear code embedding found in a computer search [i]
2. digital (213, 228, 4793484)-net over F₃, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(226, 226+15, large)-Net over F₃ — Digital

Digital (226, 241, large)-net over F₃, using

t-expansion [i] based on digital (221, 241, large)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, large, F₃, 20) (dual of [large, large−241, 21]-code), using
  - strength reduction [i] based on linear OA(3²⁴¹, large, F₃, 25) (dual of [large, large−241, 26]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(226, 226+15, large)-Net in Base 3 — Upper bound on s

There is no (226, 241, large)-net in base 3, because

13 times m-reduction [i] would yield (226, 228, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]