MinT - Best Known (82−35, 82, s)-Nets in Base 64

Best Known (82−35, 82, s)-Nets in Base 64

(82−35, 82, 593)-Net over F₆₄ — Constructive and digital

Digital (47, 82, 593)-net over F₆₄, using

1 times m-reduction [i] based on digital (47, 83, 593)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 19, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, 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]

(82−35, 82, 964)-Net in Base 64 — Constructive

(47, 82, 964)-net in base 64, using

net defined by OOA [i] based on OOA(64⁸², 964, S₆₄, 35, 35), using
- OOA 17-folding and stacking with additional row [i] based on OA(64⁸², 16389, S₆₄, 35), using
  - discarding factors based on OA(64⁸², 16390, S₆₄, 35), using
    - discarding parts of the base [i] based on linear OA(128⁷⁰, 16390, F₁₂₈, 35) (dual of [16390, 16320, 36]-code), using
      - constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
        linear OA(128⁶⁹, 16385, F₁₂₈, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
        linear OA(128⁶⁵, 16385, F₁₂₈, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
        linear OA(128¹, 5, F₁₂₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(82−35, 82, 4974)-Net over F₆₄ — Digital

Digital (47, 82, 4974)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸², 4974, F₆₄, 35) (dual of [4974, 4892, 36]-code), using
- 863 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 31 times 0, 1, 62 times 0, 1, 122 times 0, 1, 230 times 0, 1, 390 times 0) [i] based on linear OA(64⁶⁹, 4098, F₆₄, 35) (dual of [4098, 4029, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. 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]
    2. linear OA(64⁶⁷, 4096, F₆₄, 34) (dual of [4096, 4029, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(82−35, 82, large)-Net in Base 64 — Upper bound on s

There is no (47, 82, large)-net in base 64, because

33 times m-reduction [i] would yield (47, 49, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]