MinT - Best Known (86−44, 86, s)-Nets in Base 64

Best Known (86−44, 86, s)-Nets in Base 64

(86−44, 86, 513)-Net over F₆₄ — Constructive and digital

Digital (42, 86, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 86, 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]

(86−44, 86, 1166)-Net over F₆₄ — Digital

Digital (42, 86, 1166)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁶, 1166, F₆₄, 44) (dual of [1166, 1080, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁸⁶, 1369, F₆₄, 44) (dual of [1369, 1283, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(41) [i] based on
    1. linear OA(64⁸⁵, 1366, F₆₄, 44) (dual of [1366, 1281, 45]-code), using an extension C_e(43) of the narrow-sense BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(64⁸³, 1366, F₆₄, 42) (dual of [1366, 1283, 43]-code), using an extension C_e(41) of the narrow-sense BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    3. linear OA(64¹, 3, F₆₄, 1) (dual of [3, 2, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(86−44, 86, 1652081)-Net in Base 64 — Upper bound on s

There is no (42, 86, 1652082)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 214527 724932 355531 517285 714809 031706 489201 874554 353030 524527 463161 838078 062517 433798 414612 717117 964972 433094 987950 127859 646584 650922 515862 169279 171032 485680 > 64⁸⁶ [i]