MinT - Best Known (87−44, 87, s)-Nets in Base 64

Best Known (87−44, 87, s)-Nets in Base 64

(87−44, 87, 513)-Net over F₆₄ — Constructive and digital

Digital (43, 87, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 87, 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]

(87−44, 87, 1403)-Net over F₆₄ — Digital

Digital (43, 87, 1403)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁸⁷, 1403, F₆₄, 2, 44) (dual of [(1403, 2), 2719, 45]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁸⁷, 2049, F₆₄, 2, 44) (dual of [(2049, 2), 4011, 45]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁸⁷, 4098, F₆₄, 44) (dual of [4098, 4011, 45]-code), using
    - constructionÂ X applied to C_e(43) âŠ‚ C_e(42) [i] based on
      1. 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]
      2. 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]
      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]

(87−44, 87, 1995863)-Net in Base 64 — Upper bound on s

There is no (43, 87, 1995864)-net in base 64, because

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]