MinT - Best Known (123−25, 123, s)-Nets in Base 8

Best Known (123−25, 123, s)-Nets in Base 8

(123−25, 123, 2758)-Net over F₈ — Constructive and digital

Digital (98, 123, 2758)-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 (81, 106, 2730)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁰⁶, 2730, F₈, 25, 25) (dual of [(2730, 25), 68144, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(8¹⁰⁶, 32761, F₈, 25) (dual of [32761, 32655, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁰⁶, 32768, F₈, 25) (dual of [32768, 32662, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(123−25, 123, 59523)-Net over F₈ — Digital

Digital (98, 123, 59523)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−25, 123, large)-Net in Base 8 — Upper bound on s

There is no (98, 123, large)-net in base 8, because

23 times m-reduction [i] would yield (98, 100, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]