MinT - Best Known (144−28, 144, s)-Nets in Base 3

Best Known (144−28, 144, s)-Nets in Base 3

(144−28, 144, 688)-Net over F₃ — Constructive and digital

Digital (116, 144, 688)-net over F₃, using

t-expansion [i] based on digital (115, 144, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 36, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(144−28, 144, 2196)-Net over F₃ — Digital

Digital (116, 144, 2196)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁴, 2196, F₃, 28) (dual of [2196, 2052, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁴, 3281, F₃, 28) (dual of [3281, 3137, 29]-code), using
  - the BCH-code C(I) with length 3281Â | 3¹⁶âˆ’1, defining interval IÂ =Â {−27,−25,−23,…,27}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(144−28, 144, 244307)-Net in Base 3 — Upper bound on s

There is no (116, 144, 244308)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 507 545736 735165 195184 156400 794198 399920 997756 348333 241885 098480 551609 > 3¹⁴⁴ [i]