MinT - Best Known (85−49, 85, s)-Nets in Base 32

Best Known (85−49, 85, s)-Nets in Base 32

(85−49, 85, 174)-Net over F₃₂ — Constructive and digital

Digital (36, 85, 174)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 29, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 56, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(85−49, 85, 278)-Net over F₃₂ — Digital

Digital (36, 85, 278)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁸⁵, 278, F₃₂, 2, 49) (dual of [(278, 2), 471, 50]-NRT-code), using
- constructionÂ X applied to AG(2;F,494P) âŠ‚ AG(2;F,501P) [i] based on
  1. linear OOA(32⁷⁹, 272, F₃₂, 2, 49) (dual of [(272, 2), 465, 50]-NRT-code), using algebraic-geometric NRT-code AG(2;F,494P) [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]
  2. linear OOA(32⁷², 272, F₃₂, 2, 42) (dual of [(272, 2), 472, 43]-NRT-code), using algebraic-geometric NRT-code AG(2;F,501P) [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273 (see above)
  3. linear OOA(32⁶, 6, F₃₂, 2, 6) (dual of [(6, 2), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(32⁶, 32, F₃₂, 2, 6) (dual of [(32, 2), 58, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;58,32) [i]

(85−49, 85, 288)-Net in Base 32 — Constructive

(36, 85, 288)-net in base 32, using

t-expansion [i] based on (35, 85, 288)-net in base 32, using
- 6 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
  - base change [i] based on digital (9, 65, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(85−49, 85, 342)-Net in Base 32

(36, 85, 342)-net in base 32, using

11 times m-reduction [i] based on (36, 96, 342)-net in base 32, using
- base change [i] based on digital (20, 80, 342)-net over F₆₄, using
  - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
      - a curve from the manYPoints database [i]

(85−49, 85, 58605)-Net in Base 32 — Upper bound on s

There is no (36, 85, 58606)-net in base 32, because

1 times m-reduction [i] would yield (36, 84, 58606)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 708741 454089 461266 245371 382031 531583 581425 370132 585111 904017 502479 081637 217454 331103 775236 802443 182891 897853 198337 749158 668938 > 32⁸⁴ [i]