MinT - Best Known (48, 48+53, s)-Nets in Base 8

Best Known (48, 48+53, s)-Nets in Base 8

(48, 48+53, 100)-Net over F₈ — Constructive and digital

Digital (48, 101, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 34, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [i]
2. digital (14, 67, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(48, 48+53, 145)-Net over F₈ — Digital

Digital (48, 101, 145)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8¹⁰¹, 145, F₈, 3, 53) (dual of [(145, 3), 334, 54]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8⁹⁸, 144, F₈, 3, 53) (dual of [(144, 3), 334, 54]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(3;F,378P) [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(48, 48+53, 4467)-Net in Base 8 — Upper bound on s

There is no (48, 101, 4468)-net in base 8, because

1 times m-reduction [i] would yield (48, 100, 4468)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 045171 807236 422927 232172 079189 468180 226493 142970 361350 373632 172958 887875 948215 091195 611868 > 8¹⁰⁰ [i]