MinT - Best Known (52, 71, s)-Nets in Base 4

Best Known (52, 71, s)-Nets in Base 4

(52, 71, 312)-Net over F₄ — Constructive and digital

Digital (52, 71, 312)-net over F₄, using

t-expansion [i] based on digital (51, 71, 312)-net over F₄, using
- 1 times m-reduction [i] based on digital (51, 72, 312)-net over F₄, using
  - trace code for nets [i] based on digital (3, 24, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(52, 71, 387)-Net in Base 4 — Constructive

(52, 71, 387)-net in base 4, using

1 times m-reduction [i] based on (52, 72, 387)-net in base 4, using
- trace code for nets [i] based on (4, 24, 129)-net in base 64, using
  - 4 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
    - base change [i] based on digital (0, 24, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]

(52, 71, 708)-Net over F₄ — Digital

Digital (52, 71, 708)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷¹, 708, F₄, 19) (dual of [708, 637, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁷¹, 1023, F₄, 19) (dual of [1023, 952, 20]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(52, 71, 66569)-Net in Base 4 — Upper bound on s

There is no (52, 71, 66570)-net in base 4, because

1 times m-reduction [i] would yield (52, 70, 66570)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 393946 923664 386366 475604 988760 461603 423783 > 4⁷⁰ [i]