MinT - Best Known (62, 77, s)-Nets in Base 4

Best Known (62, 77, s)-Nets in Base 4

(62, 77, 1076)-Net over F₄ — Constructive and digital

Digital (62, 77, 1076)-net over F₄, using

4¹ times duplication [i] based on digital (61, 76, 1076)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 16, 48)-net over F₄, using
    - trace code for nets [i] based on digital (1, 8, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (45, 60, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 15, 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]

(62, 77, 5461)-Net over F₄ — Digital

Digital (62, 77, 5461)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁷, 5461, F₄, 15) (dual of [5461, 5384, 16]-code), using
- the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,4,7,…,43}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(62, 77, 3876574)-Net in Base 4 — Upper bound on s

There is no (62, 77, 3876575)-net in base 4, because

1 times m-reduction [i] would yield (62, 76, 3876575)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5708 996329 133524 689604 808297 881690 588742 399596 > 4⁷⁶ [i]