MinT - Best Known (77−39, 77, s)-Nets in Base 32

Best Known (77−39, 77, s)-Nets in Base 32

(77−39, 77, 218)-Net over F₃₂ — Constructive and digital

Digital (38, 77, 218)-net over F₃₂, using

1 times m-reduction [i] based on digital (38, 78, 218)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 27, 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, 51, 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]

(77−39, 77, 288)-Net in Base 32 — Constructive

(38, 77, 288)-net in base 32, using

t-expansion [i] based on (37, 77, 288)-net in base 32, using
- 21 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]

(77−39, 77, 567)-Net over F₃₂ — Digital

Digital (38, 77, 567)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁷, 567, F₃₂, 39) (dual of [567, 490, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷⁷, 1025, F₃₂, 39) (dual of [1025, 948, 40]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]

(77−39, 77, 268187)-Net in Base 32 — Upper bound on s

There is no (38, 77, 268188)-net in base 32, because

1 times m-reduction [i] would yield (38, 76, 268188)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462752 751426 789051 725871 001598 727656 596398 836204 286505 721166 047199 098166 796986 205234 237185 634545 653058 742986 494717 > 32⁷⁶ [i]