MinT - Best Known (79−40, 79, s)-Nets in Base 32

Best Known (79−40, 79, s)-Nets in Base 32

(79−40, 79, 218)-Net over F₃₂ — Constructive and digital

Digital (39, 79, 218)-net over F₃₂, using

2 times m-reduction [i] based on digital (39, 81, 218)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 28, 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 (11, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(79−40, 79, 288)-Net in Base 32 — Constructive

(39, 79, 288)-net in base 32, using

26 times m-reduction [i] based on (39, 105, 288)-net in base 32, using
- base change [i] based on digital (9, 75, 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]

(79−40, 79, 579)-Net over F₃₂ — Digital

Digital (39, 79, 579)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁹, 579, F₃₂, 40) (dual of [579, 500, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷⁹, 1035, F₃₂, 40) (dual of [1035, 956, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(35) [i] based on
    1. linear OA(32⁷⁶, 1024, F₃₂, 40) (dual of [1024, 948, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(32⁶⁸, 1024, F₃₂, 36) (dual of [1024, 956, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    3. linear OA(32³, 11, F₃₂, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
      - discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]

(79−40, 79, 236193)-Net in Base 32 — Upper bound on s

There is no (39, 79, 236194)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 80697 401855 215511 511014 654712 808782 744809 823041 655268 128045 872621 171875 540544 031290 655135 131834 147182 490656 694229 652552 > 32⁷⁹ [i]