MinT - Best Known (98−19, 98, s)-Nets in Base 4

Best Known (98−19, 98, s)-Nets in Base 4

(98−19, 98, 1094)-Net over F₄ — Constructive and digital

Digital (79, 98, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 22, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 11, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (57, 76, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 19, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(98−19, 98, 5461)-Net over F₄ — Digital

Digital (79, 98, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁸, 5461, F₄, 19) (dual of [5461, 5363, 20]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,55}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(98−19, 98, 4260869)-Net in Base 4 — Upper bound on s

There is no (79, 98, 4260870)-net in base 4, because

1 times m-reduction [i] would yield (79, 97, 4260870)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 25108 411453 822682 557354 276902 352181 039596 600685 963468 975488 > 4⁹⁷ [i]