MinT - Best Known (4, 4+5, s)-Nets in Base 8

Best Known (4, 4+5, s)-Nets in Base 8

(4, 4+5, 65)-Net over F₈ — Constructive and digital

Digital (4, 9, 65)-net over F₈, using

base reduction for projective spaces (embedding PG(4,64) in PG(8,8)) for nets [i] based on digital (0, 5, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(4, 4+5, 66)-Net over F₈ — Digital

Digital (4, 9, 66)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹, 66, F₈, 5) (dual of [66, 57, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(8², 9, F₈, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
    - extended Reedâ€“Solomon code RS_e(7,8) [i]
    - Hamming code H(2,8) [i]
  2. linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
    - a code by Danev and Olsson [i]

(4, 4+5, 826)-Net in Base 8 — Upper bound on s

There is no (4, 9, 827)-net in base 8, because

1 times m-reduction [i] would yield (4, 8, 827)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 16 788101 > 8⁸ [i]