MinT - Best Known (158−21, 158, s)-Nets in Base 8

Best Known (158−21, 158, s)-Nets in Base 8

(158−21, 158, 838884)-Net over F₈ — Constructive and digital

Digital (137, 158, 838884)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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 (124, 145, 838860)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁵, 838860, F₈, 21, 21) (dual of [(838860, 21), 17615915, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹⁴⁵, 8388601, F₈, 21) (dual of [8388601, 8388456, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(158−21, 158, large)-Net over F₈ — Digital

Digital (137, 158, large)-net over F₈, using

8⁵ times duplication [i] based on digital (132, 153, large)-net over F₈, using
- t-expansion [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]

(158−21, 158, large)-Net in Base 8 — Upper bound on s

There is no (137, 158, large)-net in base 8, because

19 times m-reduction [i] would yield (137, 139, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]