MinT - Best Known (98, 98+27, s)-Nets in Base 3

Best Known (98, 98+27, s)-Nets in Base 3

(98, 98+27, 600)-Net over F₃ — Constructive and digital

Digital (98, 125, 600)-net over F₃, using

3¹ times duplication [i] based on digital (97, 124, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 31, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(98, 98+27, 1092)-Net over F₃ — Digital

Digital (98, 125, 1092)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁵, 1092, F₃, 27) (dual of [1092, 967, 28]-code), using
- 1 times truncation [i] based on linear OA(3¹²⁶, 1093, F₃, 28) (dual of [1093, 967, 29]-code), using
  - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,57,113,…,420}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(98, 98+27, 100768)-Net in Base 3 — Upper bound on s

There is no (98, 125, 100769)-net in base 3, because

1 times m-reduction [i] would yield (98, 124, 100769)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 145566 092585 539146 482421 591898 011115 969678 885648 814062 274835 > 3¹²⁴ [i]