MinT - Best Known (124, 124+31, s)-Nets in Base 8

Best Known (124, 124+31, s)-Nets in Base 8

(124, 124+31, 2209)-Net over F₈ — Constructive and digital

Digital (124, 155, 2209)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (105, 136, 2184)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁶, 2184, F₈, 31, 31) (dual of [(2184, 31), 67568, 32]-NRT-code), using
    - OOA 15-folding and stacking with additional row [i] based on linear OA(8¹³⁶, 32761, F₈, 31) (dual of [32761, 32625, 32]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹³⁶, 32768, F₈, 31) (dual of [32768, 32632, 32]-code), using
        an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(124, 124+31, 79751)-Net over F₈ — Digital

Digital (124, 155, 79751)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+31, large)-Net in Base 8 — Upper bound on s

There is no (124, 155, large)-net in base 8, because

29 times m-reduction [i] would yield (124, 126, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]