MinT - Best Known (65−36, 65, s)-Nets in Base 64

Best Known (65−36, 65, s)-Nets in Base 64

(65−36, 65, 513)-Net over F₆₄ — Constructive and digital

Digital (29, 65, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 65, 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]

(65−36, 65, 525)-Net over F₆₄ — Digital

Digital (29, 65, 525)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁶⁵, 525, F₆₄, 36) (dual of [525, 460, 37]-code), using
- 11 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 10 times 0) [i] based on linear OA(64⁶⁴, 513, F₆₄, 36) (dual of [513, 449, 37]-code), using
  - extended algebraic-geometric code AG_e(F,476P) [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(65−36, 65, 399113)-Net in Base 64 — Upper bound on s

There is no (29, 65, 399114)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2521 774609 085566 533227 890246 813810 030303 749653 641165 305030 123086 932426 386189 870217 463529 506368 392064 814893 619497 956160 > 64⁶⁵ [i]