MinT - Best Known (51, 51+46, s)-Nets in Base 32

Best Known (51, 51+46, s)-Nets in Base 32

(51, 51+46, 240)-Net over F₃₂ — Constructive and digital

Digital (51, 97, 240)-net over F₃₂, using

12 times m-reduction [i] based on digital (51, 109, 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, 69, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(51, 51+46, 513)-Net in Base 32 — Constructive

(51, 97, 513)-net in base 32, using

t-expansion [i] based on (46, 97, 513)-net in base 32, using
- 11 times m-reduction [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]

(51, 51+46, 1050)-Net over F₃₂ — Digital

Digital (51, 97, 1050)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁷, 1050, F₃₂, 46) (dual of [1050, 953, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁷, 1053, F₃₂, 46) (dual of [1053, 956, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(35) [i] based on
    1. linear OA(32⁸⁸, 1024, F₃₂, 46) (dual of [1024, 936, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    2. linear OA(32⁶⁸, 1024, F₃₂, 36) (dual of [1024, 956, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    3. linear OA(32⁹, 29, F₃₂, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁹, 32, F₃₂, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
        Reedâ€“Solomon code RS(23,32) [i]

(51, 51+46, 677475)-Net in Base 32 — Upper bound on s

There is no (51, 97, 677476)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 99 898206 041866 388333 576538 265117 316099 808906 243384 808682 279055 592454 041897 816855 795740 862715 240067 151629 139778 354027 916412 895322 610518 811843 700888 > 32⁹⁷ [i]