MinT - Best Known (117−18, 117, s)-Nets in Base 16

Best Known (117−18, 117, s)-Nets in Base 16

(117−18, 117, 1864179)-Net over F₁₆ — Constructive and digital

Digital (99, 117, 1864179)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (86, 104, 1864134)-net over F₁₆, using
  - trace code for nets [i] based on digital (34, 52, 932067)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁵², 932067, F₂₅₆, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(256⁵², large, F₂₅₆, 18) (dual of [large, large−52, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(117−18, 117, large)-Net over F₁₆ — Digital

Digital (99, 117, large)-net over F₁₆, using

4 times m-reduction [i] based on digital (99, 121, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²¹, large, F₁₆, 22) (dual of [large, large−121, 23]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(117−18, 117, large)-Net in Base 16 — Upper bound on s

There is no (99, 117, large)-net in base 16, because

16 times m-reduction [i] would yield (99, 101, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]