MinT - Best Known (217−31, 217, s)-Nets in Base 3

Best Known (217−31, 217, s)-Nets in Base 3

(217−31, 217, 3943)-Net over F₃ — Constructive and digital

Digital (186, 217, 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, 201, 3936)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁰¹, 3936, F₃, 31, 31) (dual of [(3936, 31), 121815, 32]-NRT-code), using
    - OOA 15-folding and stacking with additional row [i] based on linear OA(3²⁰¹, 59041, F₃, 31) (dual of [59041, 58840, 32]-code), using
      - discarding factors / shortening the dual code based on 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]

(217−31, 217, 26010)-Net over F₃ — Digital

Digital (186, 217, 26010)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁷, 26010, F₃, 2, 31) (dual of [(26010, 2), 51803, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁷, 29557, F₃, 2, 31) (dual of [(29557, 2), 58897, 32]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²¹⁷, 59114, F₃, 31) (dual of [59114, 58897, 32]-code), using
    - discarding factors / shortening the dual code 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
        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]
        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]
        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]

(217−31, 217, large)-Net in Base 3 — Upper bound on s

There is no (186, 217, large)-net in base 3, because

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