MinT - Best Known (120−21, 120, s)-Nets in Base 8

Best Known (120−21, 120, s)-Nets in Base 8

(120−21, 120, 26228)-Net over F₈ — Constructive and digital

Digital (99, 120, 26228)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (88, 109, 26214)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁰⁹, 26214, F₈, 21, 21) (dual of [(26214, 21), 550385, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(8¹⁰⁹, 262141, F₈, 21) (dual of [262141, 262032, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁰⁹, 262144, F₈, 21) (dual of [262144, 262035, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(120−21, 120, 311001)-Net over F₈ — Digital

Digital (99, 120, 311001)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(120−21, 120, large)-Net in Base 8 — Upper bound on s

There is no (99, 120, large)-net in base 8, because

19 times m-reduction [i] would yield (99, 101, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]