MinT - Best Known (24−11, 24, s)-Nets in Base 32

Best Known (24−11, 24, s)-Nets in Base 32

(24−11, 24, 207)-Net over F₃₂ — Constructive and digital

Digital (13, 24, 207)-net over F₃₂, using

net defined by OOA [i] based on linear OOA(32²⁴, 207, F₃₂, 11, 11) (dual of [(207, 11), 2253, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(32²⁴, 1036, F₃₂, 11) (dual of [1036, 1012, 12]-code), using
  - constructionÂ X applied to C([0,5]) âŠ‚ C([0,3]) [i] based on
    1. linear OA(32²¹, 1025, F₃₂, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
    2. linear OA(32¹³, 1025, F₃₂, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (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]

(24−11, 24, 322)-Net in Base 32 — Constructive

(13, 24, 322)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (1, 6, 65)-net in base 32, using
  - base change [i] based on digital (0, 5, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. (7, 18, 257)-net in base 32, using
  - base change [i] based on (4, 15, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
      - base change [i] based on digital (0, 12, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(24−11, 24, 936)-Net over F₃₂ — Digital

Digital (13, 24, 936)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²⁴, 936, F₃₂, 11) (dual of [936, 912, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(32²⁴, 1036, F₃₂, 11) (dual of [1036, 1012, 12]-code), using
  - constructionÂ X applied to C([0,5]) âŠ‚ C([0,3]) [i] based on
    1. linear OA(32²¹, 1025, F₃₂, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
    2. linear OA(32¹³, 1025, F₃₂, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (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]

(24−11, 24, 704957)-Net in Base 32 — Upper bound on s

There is no (13, 24, 704958)-net in base 32, because

1 times m-reduction [i] would yield (13, 23, 704958)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 41538 431866 649007 149744 730382 932933 > 32²³ [i]