MinT - Best Known (76−39, 76, s)-Nets in Base 64

Best Known (76−39, 76, s)-Nets in Base 64

(76−39, 76, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 76, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 76, 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]

(76−39, 76, 1049)-Net over F₆₄ — Digital

Digital (37, 76, 1049)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁷⁶, 1049, F₆₄, 39) (dual of [1049, 973, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁷⁶, 1365, F₆₄, 39) (dual of [1365, 1289, 40]-code), using
  - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,12,23,…,419}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]

(76−39, 76, 1696408)-Net in Base 64 — Upper bound on s

There is no (37, 76, 1696409)-net in base 64, because

1 times m-reduction [i] would yield (37, 75, 1696409)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2907 380357 990015 753368 812348 629716 883028 824251 414343 022522 493963 701508 061282 611540 327598 580565 340111 142516 305767 442064 721688 718642 164152 > 64⁷⁵ [i]