MinT - Best Known (92−14, 92, s)-Nets in Base 16

Best Known (92−14, 92, s)-Nets in Base 16

(92−14, 92, 2396793)-Net over F₁₆ — Constructive and digital

Digital (78, 92, 2396793)-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 (66, 80, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(92−14, 92, 2396822)-Net in Base 16 — Constructive

(78, 92, 2396822)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (5, 12, 80)-net in base 16, using
  - base change [i] based on 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 (66, 80, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(92−14, 92, large)-Net over F₁₆ — Digital

Digital (78, 92, large)-net over F₁₆, using

t-expansion [i] based on digital (75, 92, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹², large, F₁₆, 17) (dual of [large, large−92, 18]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(16⁹¹, large, F₁₆, 17) (dual of [large, large−91, 18]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(92−14, 92, large)-Net in Base 16 — Upper bound on s

There is no (78, 92, large)-net in base 16, because

12 times m-reduction [i] would yield (78, 80, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (92−14, 92, s)-Nets in Base 16

(92−14, 92, 2396793)-Net over F16 — Constructive and digital

(92−14, 92, 2396822)-Net in Base 16 — Constructive

(92−14, 92, large)-Net over F16 — Digital

(92−14, 92, large)-Net in Base 16 — Upper bound on s

(92−14, 92, 2396793)-Net over F₁₆ — Constructive and digital

(92−14, 92, large)-Net over F₁₆ — Digital