MinT - Best Known (83−34, 83, s)-Nets in Base 64

Best Known (83−34, 83, s)-Nets in Base 64

(83−34, 83, 617)-Net over F₆₄ — Constructive and digital

Digital (49, 83, 617)-net over F₆₄, using

2 times m-reduction [i] based on 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]

(83−34, 83, 964)-Net in Base 64 — Constructive

(49, 83, 964)-net in base 64, using

1 times m-reduction [i] based on (49, 84, 964)-net in base 64, using
- base change [i] based on digital (37, 72, 964)-net over F₁₂₈, using
  - 128² times duplication [i] based on digital (35, 70, 964)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁷⁰, 964, F₁₂₈, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
      - OOA 17-folding and stacking with additional row [i] based on linear OA(128⁷⁰, 16389, F₁₂₈, 35) (dual of [16389, 16319, 36]-code), using
        discarding factors / shortening the dual code 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]

(83−34, 83, 7308)-Net over F₆₄ — Digital

Digital (49, 83, 7308)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−34, 83, large)-Net in Base 64 — Upper bound on s

There is no (49, 83, large)-net in base 64, because

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

Best Known (83−34, 83, s)-Nets in Base 64

(83−34, 83, 617)-Net over F64 — Constructive and digital

(83−34, 83, 964)-Net in Base 64 — Constructive

(83−34, 83, 7308)-Net over F64 — Digital

(83−34, 83, large)-Net in Base 64 — Upper bound on s

(83−34, 83, 617)-Net over F₆₄ — Constructive and digital

(83−34, 83, 7308)-Net over F₆₄ — Digital