MinT - Best Known (59, 59+25, s)-Nets in Base 64

Best Known (59, 59+25, s)-Nets in Base 64

(59, 59+25, 21849)-Net over F₆₄ — Constructive and digital

Digital (59, 84, 21849)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64⁸⁴, 21849, F₆₄, 25, 25) (dual of [(21849, 25), 546141, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(64⁸⁴, 262189, F₆₄, 25) (dual of [262189, 262105, 26]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁸⁴, 262192, F₆₄, 25) (dual of [262192, 262108, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(64⁷³, 262145, F₆₄, 25) (dual of [262145, 262072, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(64³⁷, 262145, F₆₄, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
      3. linear OA(64¹¹, 47, F₆₄, 11) (dual of [47, 36, 12]-code or 47-arc in PG(10,64)), using
        discarding factors / shortening the dual code based on linear OA(64¹¹, 64, F₆₄, 11) (dual of [64, 53, 12]-code or 64-arc in PG(10,64)), using
        Reedâ€“Solomon code RS(53,64) [i]

(59, 59+25, 326338)-Net over F₆₄ — Digital

Digital (59, 84, 326338)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+25, large)-Net in Base 64 — Upper bound on s

There is no (59, 84, large)-net in base 64, because

23 times m-reduction [i] would yield (59, 61, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]