MinT - Best Known (90−46, 90, s)-Nets in Base 64

Best Known (90−46, 90, s)-Nets in Base 64

(90−46, 90, 513)-Net over F₆₄ — Constructive and digital

Digital (44, 90, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 90, 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]

(90−46, 90, 1213)-Net over F₆₄ — Digital

Digital (44, 90, 1213)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁹⁰, 1213, F₆₄, 46) (dual of [1213, 1123, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁹⁰, 1371, F₆₄, 46) (dual of [1371, 1281, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(43) [i] based on
    1. linear OA(64⁸⁹, 1366, F₆₄, 46) (dual of [1366, 1277, 47]-code), using an extension C_e(45) of the narrow-sense BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [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(64¹, 5, F₆₄, 1) (dual of [5, 4, 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]

(90−46, 90, 1748935)-Net in Base 64 — Upper bound on s

There is no (44, 90, 1748936)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 599177 103036 930190 362192 761805 185419 205584 130976 957566 955244 647129 247550 186167 261479 554142 629331 527766 637968 555124 076381 496121 840923 540833 319360 116498 206818 971884 > 64⁹⁰ [i]