MinT - Best Known (88−51, 88, s)-Nets in Base 32

Best Known (88−51, 88, s)-Nets in Base 32

(88−51, 88, 174)-Net over F₃₂ — Constructive and digital

Digital (37, 88, 174)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 30, 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, 58, 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]

(88−51, 88, 277)-Net over F₃₂ — Digital

Digital (37, 88, 277)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁸⁸, 277, F₃₂, 2, 51) (dual of [(277, 2), 466, 52]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁸⁸, 279, F₃₂, 2, 51) (dual of [(279, 2), 470, 52]-NRT-code), using
  - constructionÂ X applied to AG(2;F,492P) âŠ‚ AG(2;F,500P) [i] based on
    1. linear OOA(32⁸¹, 272, F₃₂, 2, 51) (dual of [(272, 2), 463, 52]-NRT-code), using algebraic-geometric NRT-code AG(2;F,492P) [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(32⁷³, 272, F₃₂, 2, 43) (dual of [(272, 2), 471, 44]-NRT-code), using algebraic-geometric NRT-code AG(2;F,500P) [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273 (see above)
    3. linear OOA(32⁷, 7, F₃₂, 2, 7) (dual of [(7, 2), 7, 8]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(32⁷, 32, F₃₂, 2, 7) (dual of [(32, 2), 57, 8]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(2;57,32) [i]

(88−51, 88, 288)-Net in Base 32 — Constructive

(37, 88, 288)-net in base 32, using

10 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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]

(88−51, 88, 342)-Net in Base 32

(37, 88, 342)-net in base 32, using

14 times m-reduction [i] based on (37, 102, 342)-net in base 32, using
- base change [i] based on digital (20, 85, 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]

(88−51, 88, 56765)-Net in Base 32 — Upper bound on s

There is no (37, 88, 56766)-net in base 32, because

1 times m-reduction [i] would yield (37, 87, 56766)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 88726 432567 131930 303240 755466 052674 163407 869894 989492 095067 075394 583966 762365 828461 239273 066570 593783 173347 211551 070843 906540 902347 > 32⁸⁷ [i]