MinT - Best Known (220−41, 220, s)-Nets in Base 3

Best Known (220−41, 220, s)-Nets in Base 3

(220−41, 220, 896)-Net over F₃ — Constructive and digital

Digital (179, 220, 896)-net over F₃, using

t-expansion [i] based on digital (178, 220, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(220−41, 220, 3641)-Net over F₃ — Digital

Digital (179, 220, 3641)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁰, 3641, F₃, 41) (dual of [3641, 3421, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁰, 6574, F₃, 41) (dual of [6574, 6354, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(37) [i] based on
    1. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(3²⁰¹, 6561, F₃, 38) (dual of [6561, 6360, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(220−41, 220, 696212)-Net in Base 3 — Upper bound on s

There is no (179, 220, 696213)-net in base 3, because

1 times m-reduction [i] would yield (179, 219, 696213)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 717421 091985 623021 818045 467928 764584 845405 233212 242209 290261 587693 061467 921571 560417 541297 162567 123089 > 3²¹⁹ [i]