MinT - Best Known (44, 44+45, s)-Nets in Base 64

Best Known (44, 44+45, s)-Nets in Base 64

(44, 44+45, 513)-Net over F₆₄ — Constructive and digital

Digital (44, 89, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 89, 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]

(44, 44+45, 1424)-Net over F₆₄ — Digital

Digital (44, 89, 1424)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁹, 1424, F₆₄, 2, 45) (dual of [(1424, 2), 2759, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁸⁹, 2049, F₆₄, 2, 45) (dual of [(2049, 2), 4009, 46]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁸⁹, 4098, F₆₄, 45) (dual of [4098, 4009, 46]-code), using
    - constructionÂ X applied to C_e(44) âŠ‚ C_e(43) [i] based on
      1. linear OA(64⁸⁹, 4096, F₆₄, 45) (dual of [4096, 4007, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      2. linear OA(64⁸⁷, 4096, F₆₄, 44) (dual of [4096, 4009, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 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]

(44, 44+45, 2411184)-Net in Base 64 — Upper bound on s

There is no (44, 89, 2411185)-net in base 64, because

1 times m-reduction [i] would yield (44, 88, 2411185)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 878 699171 978514 958738 015003 479539 512914 598524 694539 171677 037430 132119 052181 131186 074336 829971 864220 440442 516296 962909 426125 874232 474477 494796 935046 375394 086064 > 64⁸⁸ [i]