MinT - Best Known (157−21, 157, s)-Nets in Base 8

Best Known (157−21, 157, s)-Nets in Base 8

(157−21, 157, 838877)-Net over F₈ — Constructive and digital

Digital (136, 157, 838877)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [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]

(157−21, 157, large)-Net over F₈ — Digital

Digital (136, 157, large)-net over F₈, using

8⁴ 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]

(157−21, 157, large)-Net in Base 8 — Upper bound on s

There is no (136, 157, large)-net in base 8, because

19 times m-reduction [i] would yield (136, 138, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]