MinT - Best Known (38, 38+42, s)-Nets in Base 32

Best Known (38, 38+42, s)-Nets in Base 32

(38, 38+42, 202)-Net over F₃₂ — Constructive and digital

Digital (38, 80, 202)-net over F₃₂, using

2 times m-reduction [i] based on digital (38, 82, 202)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 29, 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]
  2. digital (9, 53, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(38, 38+42, 288)-Net in Base 32 — Constructive

(38, 80, 288)-net in base 32, using

t-expansion [i] based on (37, 80, 288)-net in base 32, using
- 18 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]

(38, 38+42, 489)-Net over F₃₂ — Digital

Digital (38, 80, 489)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁸⁰, 489, F₃₂, 2, 42) (dual of [(489, 2), 898, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁸⁰, 513, F₃₂, 2, 42) (dual of [(513, 2), 946, 43]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁸⁰, 1026, F₃₂, 42) (dual of [1026, 946, 43]-code), using
    - constructionÂ X applied to C_e(41) âŠ‚ C_e(40) [i] based on
      1. linear OA(32⁸⁰, 1024, F₃₂, 42) (dual of [1024, 944, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
      2. linear OA(32⁷⁸, 1024, F₃₂, 41) (dual of [1024, 946, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      3. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(38, 38+42, 513)-Net in Base 32

(38, 80, 513)-net in base 32, using

base change [i] based on digital (8, 50, 513)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(38, 38+42, 151706)-Net in Base 32 — Upper bound on s

There is no (38, 80, 151707)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 582328 852200 222753 814690 502394 282589 200953 125655 973363 561489 504554 988251 372482 745193 571418 588094 356836 148474 503417 137324 > 32⁸⁰ [i]