MinT - Best Known (150−18, 150, s)-Nets in Base 8

Best Known (150−18, 150, s)-Nets in Base 8

(150−18, 150, 1864158)-Net over F₈ — Constructive and digital

Digital (132, 150, 1864158)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (120, 138, 1864134)-net over F₈, using
  - trace code for nets [i] based on digital (51, 69, 932067)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁹, 932067, F₆₄, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(150−18, 150, large)-Net over F₈ — Digital

Digital (132, 150, large)-net over F₈, using

t-expansion [i] based on digital (131, 150, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(150−18, 150, large)-Net in Base 8 — Upper bound on s

There is no (132, 150, large)-net in base 8, because

16 times m-reduction [i] would yield (132, 134, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]