MinT - Best Known (88−45, 88, s)-Nets in Base 64

Best Known (88−45, 88, s)-Nets in Base 64

(88−45, 88, 513)-Net over F₆₄ — Constructive and digital

Digital (43, 88, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 88, 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]

(88−45, 88, 1189)-Net over F₆₄ — Digital

Digital (43, 88, 1189)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁸, 1189, F₆₄, 45) (dual of [1189, 1101, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁸⁸, 1369, F₆₄, 45) (dual of [1369, 1281, 46]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(64⁸⁷, 1368, F₆₄, 45) (dual of [1368, 1281, 46]-code), using
    - constructionÂ X applied to C_e(44) âŠ‚ C_e(43) [i] based on
      1. linear OA(64⁸⁷, 1366, F₆₄, 45) (dual of [1366, 1279, 46]-code), using an extension C_e(44) of the narrow-sense BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      2. 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]
      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]

(88−45, 88, 1995863)-Net in Base 64 — Upper bound on s

There is no (43, 88, 1995864)-net in base 64, because

1 times m-reduction [i] would yield (43, 87, 1995864)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 729610 482149 444395 364987 990171 390582 566017 223017 971814 669317 252489 875802 722040 894767 291532 001655 822007 793573 143679 937162 189449 741286 336221 868710 370284 695355 > 64⁸⁷ [i]