MinT - Best Known (72−37, 72, s)-Nets in Base 64

Best Known (72−37, 72, s)-Nets in Base 64

(72−37, 72, 513)-Net over F₆₄ — Constructive and digital

Digital (35, 72, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 72, 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]

(72−37, 72, 1002)-Net over F₆₄ — Digital

Digital (35, 72, 1002)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁷², 1002, F₆₄, 37) (dual of [1002, 930, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁷², 1365, F₆₄, 37) (dual of [1365, 1293, 38]-code), using
  - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,24,47,…,829}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]

(72−37, 72, 1596480)-Net in Base 64 — Upper bound on s

There is no (35, 72, 1596481)-net in base 64, because

1 times m-reduction [i] would yield (35, 71, 1596481)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 173 292380 235030 994874 756173 325088 194505 958953 307021 020556 879763 407546 976573 134235 708638 785974 843985 386272 530698 558344 735360 101472 > 64⁷¹ [i]