MinT - Best Known (42, 42+43, s)-Nets in Base 64

Best Known (42, 42+43, s)-Nets in Base 64

(42, 42+43, 513)-Net over F₆₄ — Constructive and digital

Digital (42, 85, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 85, 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]

(42, 42+43, 1381)-Net over F₆₄ — Digital

Digital (42, 85, 1381)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁵, 1381, F₆₄, 2, 43) (dual of [(1381, 2), 2677, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁸⁵, 2049, F₆₄, 2, 43) (dual of [(2049, 2), 4013, 44]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁸⁵, 4098, F₆₄, 43) (dual of [4098, 4013, 44]-code), using
    - constructionÂ X applied to C_e(42) âŠ‚ C_e(41) [i] based on
      1. linear OA(64⁸⁵, 4096, F₆₄, 43) (dual of [4096, 4011, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
      2. linear OA(64⁸³, 4096, F₆₄, 42) (dual of [4096, 4013, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [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]

(42, 42+43, 2311372)-Net in Base 64 — Upper bound on s

There is no (42, 85, 2311373)-net in base 64, because

1 times m-reduction [i] would yield (42, 84, 2311373)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 52 374411 013864 001584 298517 126579 803366 098184 081014 413755 534796 830422 010351 780636 624684 340248 499500 530745 060102 387865 049925 202293 135022 769356 770192 530720 > 64⁸⁴ [i]