MinT - Best Known (64, 78, s)-Nets in Base 16

Best Known (64, 78, s)-Nets in Base 16

(64, 78, 149848)-Net over F₁₆ — Constructive and digital

Digital (64, 78, 149848)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 51)-net over F₁₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 17)-net over F₁₆, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 3, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
    3. digital (0, 7, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆ (see above)
2. digital (52, 66, 149797)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶⁶, 149797, F₁₆, 14, 14) (dual of [(149797, 14), 2097092, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(16⁶⁶, 1048579, F₁₆, 14) (dual of [1048579, 1048513, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁶⁶, 1048581, F₁₆, 14) (dual of [1048581, 1048515, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(16⁶⁶, 1048576, F₁₆, 14) (dual of [1048576, 1048510, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(16⁶¹, 1048576, F₁₆, 13) (dual of [1048576, 1048515, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(64, 78, 299617)-Net in Base 16 — Constructive

(64, 78, 299617)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. (56, 70, 299593)-net in base 16, using
  - base change [i] based on digital (26, 40, 299593)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁴⁰, 299593, F₁₂₈, 14, 14) (dual of [(299593, 14), 4194262, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(128⁴⁰, 2097151, F₁₂₈, 14) (dual of [2097151, 2097111, 15]-code), using
        discarding factors / shortening the dual code 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]

(64, 78, 6339212)-Net over F₁₆ — Digital

Digital (64, 78, 6339212)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 78, large)-Net in Base 16 — Upper bound on s

There is no (64, 78, large)-net in base 16, because

12 times m-reduction [i] would yield (64, 66, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (64, 78, s)-Nets in Base 16

(64, 78, 149848)-Net over F16 — Constructive and digital

(64, 78, 299617)-Net in Base 16 — Constructive

(64, 78, 6339212)-Net over F16 — Digital

(64, 78, large)-Net in Base 16 — Upper bound on s

(64, 78, 149848)-Net over F₁₆ — Constructive and digital

(64, 78, 6339212)-Net over F₁₆ — Digital