MinT - Best Known (137−28, 137, s)-Nets in Base 8

Best Known (137−28, 137, s)-Nets in Base 8

(137−28, 137, 2357)-Net over F₈ — Constructive and digital

Digital (109, 137, 2357)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (93, 121, 2340)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹²¹, 2340, F₈, 28, 28) (dual of [(2340, 28), 65399, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(8¹²¹, 32760, F₈, 28) (dual of [32760, 32639, 29]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹²¹, 32768, F₈, 28) (dual of [32768, 32647, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(137−28, 137, 59669)-Net over F₈ — Digital

Digital (109, 137, 59669)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−28, 137, large)-Net in Base 8 — Upper bound on s

There is no (109, 137, large)-net in base 8, because

26 times m-reduction [i] would yield (109, 111, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]