MinT - Best Known (157−32, 157, s)-Nets in Base 8

Best Known (157−32, 157, s)-Nets in Base 8

(157−32, 157, 2062)-Net over F₈ — Constructive and digital

Digital (125, 157, 2062)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (108, 140, 2048)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁰, 2048, F₈, 32, 32) (dual of [(2048, 32), 65396, 33]-NRT-code), using
    - OA 16-folding and stacking [i] based on linear OA(8¹⁴⁰, 32768, F₈, 32) (dual of [32768, 32628, 33]-code), using
      - 1 times truncation [i] based on linear OA(8¹⁴¹, 32769, F₈, 33) (dual of [32769, 32628, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 32769Â | 8¹⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(157−32, 157, 66489)-Net over F₈ — Digital

Digital (125, 157, 66489)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−32, 157, large)-Net in Base 8 — Upper bound on s

There is no (125, 157, large)-net in base 8, because

30 times m-reduction [i] would yield (125, 127, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]