MinT - Best Known (29, 29+31, s)-Nets in Base 64

Best Known (29, 29+31, s)-Nets in Base 64

(29, 29+31, 513)-Net over F₆₄ — Constructive and digital

Digital (29, 60, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 60, 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]

(29, 29+31, 863)-Net over F₆₄ — Digital

Digital (29, 60, 863)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁶⁰, 863, F₆₄, 31) (dual of [863, 803, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁶⁰, 1365, F₆₄, 31) (dual of [1365, 1305, 32]-code), using
  - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,24,47,…,691}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(29, 29+31, 1296377)-Net in Base 64 — Upper bound on s

There is no (29, 60, 1296378)-net in base 64, because

1 times m-reduction [i] would yield (29, 59, 1296378)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 36696 089552 006804 378231 138060 208233 707680 031137 642145 803595 984511 599800 781016 673490 221784 533576 468610 532344 > 64⁵⁹ [i]