MinT - Best Known (144−25, 144, s)-Nets in Base 8

Best Known (144−25, 144, s)-Nets in Base 8

(144−25, 144, 21873)-Net over F₈ — Constructive and digital

Digital (119, 144, 21873)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 17, 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 (102, 127, 21845)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²⁷, 21845, F₈, 25, 25) (dual of [(21845, 25), 545998, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(8¹²⁷, 262141, F₈, 25) (dual of [262141, 262014, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²⁷, 262144, F₈, 25) (dual of [262144, 262017, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(144−25, 144, 367129)-Net over F₈ — Digital

Digital (119, 144, 367129)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−25, 144, large)-Net in Base 8 — Upper bound on s

There is no (119, 144, large)-net in base 8, because

23 times m-reduction [i] would yield (119, 121, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]