MinT - Best Known (57, 57+53, s)-Nets in Base 32

Best Known (57, 57+53, s)-Nets in Base 32

(57, 57+53, 240)-Net over F₃₂ — Constructive and digital

Digital (57, 110, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 110, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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, 70, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(57, 57+53, 513)-Net in Base 32 — Constructive

(57, 110, 513)-net in base 32, using

32² times duplication [i] based on (55, 108, 513)-net in base 32, using
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 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]

(57, 57+53, 1032)-Net over F₃₂ — Digital

Digital (57, 110, 1032)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹¹⁰, 1032, F₃₂, 53) (dual of [1032, 922, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹¹⁰, 1042, F₃₂, 53) (dual of [1042, 932, 54]-code), using
  - constructionÂ X applied to C([0,26]) âŠ‚ C([0,23]) [i] based on
    1. linear OA(32¹⁰⁵, 1025, F₃₂, 53) (dual of [1025, 920, 54]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,26], and minimum distance dÂ â‰¥ |{−26,−25,…,26}|+1Â =Â 54 (BCH-bound) [i]
    2. linear OA(32⁹³, 1025, F₃₂, 47) (dual of [1025, 932, 48]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,23], and minimum distance dÂ â‰¥ |{−23,−22,…,23}|+1Â =Â 48 (BCH-bound) [i]
    3. linear OA(32⁵, 17, F₃₂, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁵, 32, F₃₂, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
        Reedâ€“Solomon code RS(27,32) [i]

(57, 57+53, 694984)-Net in Base 32 — Upper bound on s

There is no (57, 110, 694985)-net in base 32, because

1 times m-reduction [i] would yield (57, 109, 694985)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 176251 592032 619318 154769 965037 252916 232300 824184 870172 553272 354038 236972 965222 688935 061803 141542 764765 680418 031745 460411 380372 506376 185640 922786 435011 952215 768416 > 32¹⁰⁹ [i]