MinT - Best Known (216−30, 216, s)-Nets in Base 3

Best Known (216−30, 216, s)-Nets in Base 3

(216−30, 216, 3943)-Net over F₃ — Constructive and digital

Digital (186, 216, 3943)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 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 (170, 200, 3936)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁰⁰, 3936, F₃, 30, 30) (dual of [(3936, 30), 117880, 31]-NRT-code), using
    - OA 15-folding and stacking [i] based on linear OA(3²⁰⁰, 59040, F₃, 30) (dual of [59040, 58840, 31]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰⁰, 59049, F₃, 30) (dual of [59049, 58849, 31]-code), using
        1 times truncation [i] based on linear OA(3²⁰¹, 59050, F₃, 31) (dual of [59050, 58849, 32]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]

(216−30, 216, 29557)-Net over F₃ — Digital

Digital (186, 216, 29557)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁶, 29557, F₃, 2, 30) (dual of [(29557, 2), 58898, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²¹⁶, 59114, F₃, 30) (dual of [59114, 58898, 31]-code), using
  - 1 times truncation [i] based on linear OA(3²¹⁷, 59115, F₃, 31) (dual of [59115, 58898, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(22) [i] based on
      1. linear OA(3²⁰¹, 59049, F₃, 31) (dual of [59049, 58848, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(3¹⁶, 66, F₃, 7) (dual of [66, 50, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(216−30, 216, large)-Net in Base 3 — Upper bound on s

There is no (186, 216, large)-net in base 3, because

28 times m-reduction [i] would yield (186, 188, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]