MinT - Best Known (52, 90, s)-Nets in Base 64

Best Known (52, 90, s)-Nets in Base 64

(52, 90, 641)-Net over F₆₄ — Constructive and digital

Digital (52, 90, 641)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]
2. digital (28, 66, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(52, 90, 862)-Net in Base 64 — Constructive

(52, 90, 862)-net in base 64, using

1 times m-reduction [i] based on (52, 91, 862)-net in base 64, using
- base change [i] based on digital (39, 78, 862)-net over F₁₂₈, using
  - 128¹ times duplication [i] based on digital (38, 77, 862)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁷⁷, 862, F₁₂₈, 39, 39) (dual of [(862, 39), 33541, 40]-NRT-code), using
      - OOA 19-folding and stacking with additional row [i] based on linear OA(128⁷⁷, 16379, F₁₂₈, 39) (dual of [16379, 16302, 40]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁷⁷, 16384, F₁₂₈, 39) (dual of [16384, 16307, 40]-code), using
        an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]

(52, 90, 5773)-Net over F₆₄ — Digital

Digital (52, 90, 5773)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 90, large)-Net in Base 64 — Upper bound on s

There is no (52, 90, large)-net in base 64, because

36 times m-reduction [i] would yield (52, 54, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]