MinT - Best Known (158−18, 158, s)-Nets in Base 8

Best Known (158−18, 158, s)-Nets in Base 8

(158−18, 158, 1864294)-Net over F₈ — Constructive and digital

Digital (140, 158, 1864294)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 10, 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 (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]

(158−18, 158, large)-Net over F₈ — Digital

Digital (140, 158, large)-net over F₈, using

t-expansion [i] based on digital (138, 158, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(158−18, 158, large)-Net in Base 8 — Upper bound on s

There is no (140, 158, large)-net in base 8, because

16 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]