MinT - Best Known (167−25, 167, s)-Nets in Base 8

Best Known (167−25, 167, s)-Nets in Base 8

(167−25, 167, 174796)-Net over F₈ — Constructive and digital

Digital (142, 167, 174796)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 19, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (123, 148, 174762)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁸, 174762, F₈, 25, 25) (dual of [(174762, 25), 4368902, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(8¹⁴⁸, 2097145, F₈, 25) (dual of [2097145, 2096997, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁴⁸, 2097152, F₈, 25) (dual of [2097152, 2097004, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(167−25, 167, 2693195)-Net over F₈ — Digital

Digital (142, 167, 2693195)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−25, 167, large)-Net in Base 8 — Upper bound on s

There is no (142, 167, large)-net in base 8, because

23 times m-reduction [i] would yield (142, 144, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]