MinT - Best Known (51−14, 51, s)-Nets in Base 64

Best Known (51−14, 51, s)-Nets in Base 64

(51−14, 51, 37594)-Net over F₆₄ — Constructive and digital

Digital (37, 51, 37594)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 145)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (1, 8, 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 (26, 40, 37449)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁰, 37449, F₆₄, 14, 14) (dual of [(37449, 14), 524246, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(64⁴⁰, 262143, F₆₄, 14) (dual of [262143, 262103, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁰, 262144, F₆₄, 14) (dual of [262144, 262104, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(51−14, 51, 299595)-Net in Base 64 — Constructive

(37, 51, 299595)-net in base 64, using

net defined by OOA [i] based on OOA(64⁵¹, 299595, S₆₄, 14, 14), using
- OA 7-folding and stacking [i] based on OA(64⁵¹, 2097165, S₆₄, 14), using
  - discarding factors based on OA(64⁵¹, 2097167, S₆₄, 14), using
    - discarding parts of the base [i] based on linear OA(128⁴³, 2097167, F₁₂₈, 14) (dual of [2097167, 2097124, 15]-code), using
      - constructionÂ X applied to C_e(13) âŠ‚ C_e(9) [i] based on
        linear OA(128⁴⁰, 2097152, F₁₂₈, 14) (dual of [2097152, 2097112, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(128²⁸, 2097152, F₁₂₈, 10) (dual of [2097152, 2097124, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(128³, 15, F₁₂₈, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(51−14, 51, 1096103)-Net over F₆₄ — Digital

Digital (37, 51, 1096103)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−14, 51, large)-Net in Base 64 — Upper bound on s

There is no (37, 51, large)-net in base 64, because

12 times m-reduction [i] would yield (37, 39, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (51−14, 51, s)-Nets in Base 64

(51−14, 51, 37594)-Net over F64 — Constructive and digital

(51−14, 51, 299595)-Net in Base 64 — Constructive

(51−14, 51, 1096103)-Net over F64 — Digital

(51−14, 51, large)-Net in Base 64 — Upper bound on s

(51−14, 51, 37594)-Net over F₆₄ — Constructive and digital

(51−14, 51, 1096103)-Net over F₆₄ — Digital