MinT - Best Known (45, 45+46, s)-Nets in Base 64

Best Known (45, 45+46, s)-Nets in Base 64

(45, 45+46, 513)-Net over F₆₄ — Constructive and digital

Digital (45, 91, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 91, 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]

(45, 45+46, 1446)-Net over F₆₄ — Digital

Digital (45, 91, 1446)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁹¹, 1446, F₆₄, 2, 46) (dual of [(1446, 2), 2801, 47]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁹¹, 2049, F₆₄, 2, 46) (dual of [(2049, 2), 4007, 47]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁹¹, 4098, F₆₄, 46) (dual of [4098, 4007, 47]-code), using
    - constructionÂ X applied to C_e(45) âŠ‚ C_e(44) [i] based on
      1. linear OA(64⁹¹, 4096, F₆₄, 46) (dual of [4096, 4005, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
      2. linear OA(64⁸⁹, 4096, F₆₄, 45) (dual of [4096, 4007, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [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]

(45, 45+46, 2095577)-Net in Base 64 — Upper bound on s

There is no (45, 91, 2095578)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 230 345897 503102 038265 484871 190271 023620 824750 679689 974638 538095 083917 237402 163316 845927 525360 760347 657352 152261 338480 498449 080819 943926 606343 704417 060408 554105 960900 > 64⁹¹ [i]