MinT - Best Known (49, 77, s)-Nets in Base 64

Best Known (49, 77, s)-Nets in Base 64

(49, 77, 690)-Net over F₆₄ — Constructive and digital

Digital (49, 77, 690)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 21, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]
2. digital (28, 56, 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]

(49, 77, 4681)-Net in Base 64 — Constructive

(49, 77, 4681)-net in base 64, using

64¹ times duplication [i] based on (48, 76, 4681)-net in base 64, using
- t-expansion [i] based on (47, 76, 4681)-net in base 64, using
  - base change [i] based on digital (28, 57, 4681)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁵⁷, 4681, F₂₅₆, 29, 29) (dual of [(4681, 29), 135692, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(256⁵⁷, 65535, F₂₅₆, 29) (dual of [65535, 65478, 30]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁵⁷, 65536, F₂₅₆, 29) (dual of [65536, 65479, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(49, 77, 24562)-Net over F₆₄ — Digital

Digital (49, 77, 24562)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 77, large)-Net in Base 64 — Upper bound on s

There is no (49, 77, large)-net in base 64, because

26 times m-reduction [i] would yield (49, 51, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]