MinT - Best Known (126−28, 126, s)-Nets in Base 3

Best Known (126−28, 126, s)-Nets in Base 3

(126−28, 126, 464)-Net over F₃ — Constructive and digital

Digital (98, 126, 464)-net over F₃, using

2 times m-reduction [i] based on digital (98, 128, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 32, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(126−28, 126, 1014)-Net over F₃ — Digital

Digital (98, 126, 1014)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁶, 1014, F₃, 28) (dual of [1014, 888, 29]-code), using
- discarding factors / shortening the dual code 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]

(126−28, 126, 59486)-Net in Base 3 — Upper bound on s

There is no (98, 126, 59487)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 310086 796003 047797 637976 839604 974024 569683 078332 884370 900389 > 3¹²⁶ [i]