MinT - Best Known (65−33, 65, s)-Nets in Base 64

Best Known (65−33, 65, s)-Nets in Base 64

(65−33, 65, 513)-Net over F₆₄ — Constructive and digital

Digital (32, 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]

(65−33, 65, 1333)-Net over F₆₄ — Digital

Digital (32, 65, 1333)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁶⁵, 1333, F₆₄, 3, 33) (dual of [(1333, 3), 3934, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁶⁵, 1366, F₆₄, 3, 33) (dual of [(1366, 3), 4033, 34]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(64⁶⁵, 4098, F₆₄, 33) (dual of [4098, 4033, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(31) [i] based on
      1. linear OA(64⁶⁵, 4096, F₆₄, 33) (dual of [4096, 4031, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      2. linear OA(64⁶³, 4096, F₆₄, 32) (dual of [4096, 4033, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
      3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(65−33, 65, 1810990)-Net in Base 64 — Upper bound on s

There is no (32, 65, 1810991)-net in base 64, because

1 times m-reduction [i] would yield (32, 64, 1810991)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 39 402122 920495 570785 726784 160043 507968 769668 719779 357018 310230 449486 252673 475892 209524 208504 688260 695896 982479 923978 > 64⁶⁴ [i]