MinT - Best Known (43, 43+25, s)-Nets in Base 64

Best Known (43, 43+25, s)-Nets in Base 64

(43, 43+25, 617)-Net over F₆₄ — Constructive and digital

Digital (43, 68, 617)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 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]
2. digital (28, 53, 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]

(43, 43+25, 5461)-Net in Base 64 — Constructive

(43, 68, 5461)-net in base 64, using

base change [i] based on digital (26, 51, 5461)-net over F₂₅₆, using
- 256² times duplication [i] based on digital (24, 49, 5461)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴⁹, 5461, F₂₅₆, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(256⁴⁹, 65533, F₂₅₆, 25) (dual of [65533, 65484, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴⁹, 65536, F₂₅₆, 25) (dual of [65536, 65487, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(43, 43+25, 20407)-Net over F₆₄ — Digital

Digital (43, 68, 20407)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(43, 43+25, large)-Net in Base 64 — Upper bound on s

There is no (43, 68, large)-net in base 64, because

23 times m-reduction [i] would yield (43, 45, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]