MinT - Best Known (230−36, 230, s)-Nets in Base 3

Best Known (230−36, 230, s)-Nets in Base 3

(230−36, 230, 1492)-Net over F₃ — Constructive and digital

Digital (194, 230, 1492)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (172, 208, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 52, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(230−36, 230, 11033)-Net over F₃ — Digital

Digital (194, 230, 11033)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁰, 11033, F₃, 36) (dual of [11033, 10803, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁰, 19733, F₃, 36) (dual of [19733, 19503, 37]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3²²⁶, 19729, F₃, 36) (dual of [19729, 19503, 37]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
      1. linear OA(3²¹⁷, 19684, F₃, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. linear OA(3¹⁸¹, 19684, F₃, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      3. linear OA(3⁹, 45, F₃, 4) (dual of [45, 36, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(230−36, 230, 4716803)-Net in Base 3 — Upper bound on s

There is no (194, 230, 4716804)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 687597 841432 744109 238361 926750 002363 928612 460040 747710 344601 909377 339799 117883 135770 450093 715246 726228 210745 > 3²³⁰ [i]