MinT - Best Known (70−21, 70, s)-Nets in Base 64

Best Known (70−21, 70, s)-Nets in Base 64

(70−21, 70, 26218)-Net over F₆₄ — Constructive and digital

Digital (49, 70, 26218)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64⁷⁰, 26218, F₆₄, 21, 21) (dual of [(26218, 21), 550508, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(64⁷⁰, 262181, F₆₄, 21) (dual of [262181, 262111, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁷⁰, 262184, F₆₄, 21) (dual of [262184, 262114, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,5]) [i] based on
      1. linear OA(64⁶¹, 262145, F₆₄, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(64³¹, 262145, F₆₄, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
      3. linear OA(64⁹, 39, F₆₄, 9) (dual of [39, 30, 10]-code or 39-arc in PG(8,64)), using
        discarding factors / shortening the dual code based on linear OA(64⁹, 64, F₆₄, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
        Reedâ€“Solomon code RS(55,64) [i]

(70−21, 70, 276447)-Net over F₆₄ — Digital

Digital (49, 70, 276447)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−21, 70, large)-Net in Base 64 — Upper bound on s

There is no (49, 70, large)-net in base 64, because

19 times m-reduction [i] would yield (49, 51, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]