MinT - Best Known (58−29, 58, s)-Nets in Base 32

Best Known (58−29, 58, s)-Nets in Base 32

(58−29, 58, 196)-Net over F₃₂ — Constructive and digital

Digital (29, 58, 196)-net over F₃₂, using

1 times m-reduction [i] based on digital (29, 59, 196)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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 (7, 37, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(58−29, 58, 288)-Net in Base 32 — Constructive

(29, 58, 288)-net in base 32, using

12 times m-reduction [i] based on (29, 70, 288)-net in base 32, using
- base change [i] based on digital (9, 50, 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]

(58−29, 58, 518)-Net over F₃₂ — Digital

Digital (29, 58, 518)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁵⁸, 518, F₃₂, 29) (dual of [518, 460, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁵⁸, 1023, F₃₂, 29) (dual of [1023, 965, 30]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {2,3,…,30}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(58−29, 58, 261935)-Net in Base 32 — Upper bound on s

There is no (29, 58, 261936)-net in base 32, because

1 times m-reduction [i] would yield (29, 57, 261936)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 62 165495 161862 757573 031512 722476 871755 091192 674114 915385 766755 425473 935246 613918 740843 > 32⁵⁷ [i]