MinT - Best Known (82−36, 82, s)-Nets in Base 64

Best Known (82−36, 82, s)-Nets in Base 64

(82−36, 82, 578)-Net over F₆₄ — Constructive and digital

Digital (46, 82, 578)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (28, 64, 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]

(82−36, 82, 4300)-Net over F₆₄ — Digital

Digital (46, 82, 4300)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸², 4300, F₆₄, 36) (dual of [4300, 4218, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁸², 4323, F₆₄, 36) (dual of [4323, 4241, 37]-code), using
  - 214 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 0, 1, 4 times 0, 1, 13 times 0, 1, 26 times 0, 1, 55 times 0, 1, 108 times 0) [i] based on linear OA(64⁷¹, 4098, F₆₄, 36) (dual of [4098, 4027, 37]-code), using
    - constructionÂ X applied to C_e(35) âŠ‚ C_e(34) [i] based on
      1. linear OA(64⁷¹, 4096, F₆₄, 36) (dual of [4096, 4025, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
      2. linear OA(64⁶⁹, 4096, F₆₄, 35) (dual of [4096, 4027, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [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]

(82−36, 82, large)-Net in Base 64 — Upper bound on s

There is no (46, 82, large)-net in base 64, because

34 times m-reduction [i] would yield (46, 48, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]