MinT - Best Known (53, 53+62, s)-Nets in Base 8

Best Known (53, 53+62, s)-Nets in Base 8

(53, 53+62, 100)-Net over F₈ — Constructive and digital

Digital (53, 115, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 39, 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, 76, 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]

(53, 53+62, 145)-Net over F₈ — Digital

Digital (53, 115, 145)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8¹¹⁵, 145, F₈, 3, 62) (dual of [(145, 3), 320, 63]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8¹¹⁵, 148, F₈, 3, 62) (dual of [(148, 3), 329, 63]-NRT-code), using
  - strength reduction [i] based on linear OOA(8¹¹⁵, 148, F₈, 3, 63) (dual of [(148, 3), 329, 64]-NRT-code), using
    - constructionÂ X applied to AG(3;F,365P) âŠ‚ AG(3;F,373P) [i] based on
      1. linear OOA(8¹⁰⁸, 143, F₈, 3, 63) (dual of [(143, 3), 321, 64]-NRT-code), using algebraic-geometric NRT-code AG(3;F,365P) [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
        Drinfeld modules of rank 1 [i]
      2. linear OOA(8¹⁰⁰, 143, F₈, 3, 55) (dual of [(143, 3), 329, 56]-NRT-code), using algebraic-geometric NRT-code AG(3;F,373P) [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144 (see above)
      3. linear OOA(8⁷, 5, F₈, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8⁷, 8, F₈, 3, 7) (dual of [(8, 3), 17, 8]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;17,8) [i]

(53, 53+62, 3953)-Net in Base 8 — Upper bound on s

There is no (53, 115, 3954)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 952608 610923 792085 197250 045251 442841 155339 117176 949851 365125 945830 709559 552910 366395 920806 224500 395888 > 8¹¹⁵ [i]