MinT - Best Known (113−18, 113, s)-Nets in Base 16

Best Known (113−18, 113, s)-Nets in Base 16

(113−18, 113, 1864151)-Net over F₁₆ — Constructive and digital

Digital (95, 113, 1864151)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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]
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]

(113−18, 113, large)-Net over F₁₆ — Digital

Digital (95, 113, large)-net over F₁₆, using

t-expansion [i] based on digital (94, 113, large)-net over F₁₆, using
- 2 times m-reduction [i] based on digital (94, 115, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁵, large, F₁₆, 21) (dual of [large, large−115, 22]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(113−18, 113, large)-Net in Base 16 — Upper bound on s

There is no (95, 113, large)-net in base 16, because

16 times m-reduction [i] would yield (95, 97, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]