MinT - Best Known (43, 43+41, s)-Nets in Base 32

Best Known (43, 43+41, s)-Nets in Base 32

(43, 43+41, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 84, 240)-net over F₃₂, using

1 times m-reduction [i] based on digital (43, 85, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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]
  2. digital (11, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(43, 43+41, 513)-Net in Base 32 — Constructive

(43, 84, 513)-net in base 32, using

6 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(43, 43+41, 775)-Net over F₃₂ — Digital

Digital (43, 84, 775)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁴, 775, F₃₂, 41) (dual of [775, 691, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁴, 1036, F₃₂, 41) (dual of [1036, 952, 42]-code), using
  - constructionÂ X applied to C([0,20]) âŠ‚ C([0,18]) [i] based on
    1. linear OA(32⁸¹, 1025, F₃₂, 41) (dual of [1025, 944, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    2. linear OA(32⁷³, 1025, F₃₂, 37) (dual of [1025, 952, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [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]

(43, 43+41, 472397)-Net in Base 32 — Upper bound on s

There is no (43, 84, 472398)-net in base 32, because

1 times m-reduction [i] would yield (43, 83, 472398)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 84616 839348 128450 133930 050114 808428 265805 646902 966966 128065 069051 161266 524643 106000 756306 389579 967384 186813 392717 012973 695196 > 32⁸³ [i]