MinT - Best Known (91−34, 91, s)-Nets in Base 64

Best Known (91−34, 91, s)-Nets in Base 64

(91−34, 91, 697)-Net over F₆₄ — Constructive and digital

Digital (57, 91, 697)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 29, 184)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 9, 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 (3, 20, 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, 62, 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]

(91−34, 91, 3855)-Net in Base 64 — Constructive

(57, 91, 3855)-net in base 64, using

64¹ times duplication [i] based on (56, 90, 3855)-net in base 64, using
- net defined by OOA [i] based on OOA(64⁹⁰, 3855, S₆₄, 34, 34), using
  - OA 17-folding and stacking [i] based on OA(64⁹⁰, 65535, S₆₄, 34), using
    - discarding factors based on OA(64⁹⁰, 65538, S₆₄, 34), using
      - discarding parts of the base [i] based on linear OA(256⁶⁷, 65538, F₂₅₆, 34) (dual of [65538, 65471, 35]-code), using
        constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
        linear OA(256⁶⁷, 65536, F₂₅₆, 34) (dual of [65536, 65469, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
        linear OA(256⁶⁵, 65536, F₂₅₆, 33) (dual of [65536, 65471, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(91−34, 91, 20001)-Net over F₆₄ — Digital

Digital (57, 91, 20001)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−34, 91, large)-Net in Base 64 — Upper bound on s

There is no (57, 91, large)-net in base 64, because

32 times m-reduction [i] would yield (57, 59, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]