MinT - Best Known (85−36, 85, s)-Nets in Base 64

Best Known (85−36, 85, s)-Nets in Base 64

(85−36, 85, 617)-Net over F₆₄ — Constructive and digital

Digital (49, 85, 617)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [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]

(85−36, 85, 910)-Net in Base 64 — Constructive

(49, 85, 910)-net in base 64, using

1 times m-reduction [i] based on (49, 86, 910)-net in base 64, using
- net defined by OOA [i] based on OOA(64⁸⁶, 910, S₆₄, 37, 37), using
  - OOA 18-folding and stacking with additional row [i] based on OA(64⁸⁶, 16381, S₆₄, 37), using
    - discarding factors based on OA(64⁸⁶, 16386, S₆₄, 37), using
      - discarding parts of the base [i] based on linear OA(128⁷³, 16386, F₁₂₈, 37) (dual of [16386, 16313, 38]-code), using
        constructionÂ X applied to C_e(36) âŠ‚ C_e(35) [i] based on
        linear OA(128⁷³, 16384, F₁₂₈, 37) (dual of [16384, 16311, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
        linear OA(128⁷¹, 16384, F₁₂₈, 36) (dual of [16384, 16313, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(85−36, 85, 5411)-Net over F₆₄ — Digital

Digital (49, 85, 5411)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁵, 5411, F₆₄, 36) (dual of [5411, 5326, 37]-code), using
- 1299 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, 1, 204 times 0, 1, 353 times 0, 1, 525 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]

(85−36, 85, large)-Net in Base 64 — Upper bound on s

There is no (49, 85, large)-net in base 64, because

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