MinT - Best Known (152−20, 152, s)-Nets in Base 8

Best Known (152−20, 152, s)-Nets in Base 8

(152−20, 152, 838888)-Net over F₈ — Constructive and digital

Digital (132, 152, 838888)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (117, 137, 838860)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁷, 838860, F₈, 20, 20) (dual of [(838860, 20), 16777063, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(8¹³⁷, 8388600, F₈, 20) (dual of [8388600, 8388463, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹³⁷, large, F₈, 20) (dual of [large, large−137, 21]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(152−20, 152, large)-Net over F₈ — Digital

Digital (132, 152, large)-net over F₈, using

t-expansion [i] based on digital (131, 152, large)-net over F₈, using
- 1 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]

(152−20, 152, large)-Net in Base 8 — Upper bound on s

There is no (132, 152, large)-net in base 8, because

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