MinT - Best Known (105−51, 105, s)-Nets in Base 32

Best Known (105−51, 105, s)-Nets in Base 32

(105−51, 105, 240)-Net over F₃₂ — Constructive and digital

Digital (54, 105, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 105, 240)-net over F₃₂, using
- 4 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)

(105−51, 105, 513)-Net in Base 32 — Constructive

(54, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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]

(105−51, 105, 942)-Net over F₃₂ — Digital

Digital (54, 105, 942)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁵, 942, F₃₂, 51) (dual of [942, 837, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁵, 1047, F₃₂, 51) (dual of [1047, 942, 52]-code), using
  - constructionÂ X applied to C_e(50) âŠ‚ C_e(42) [i] based on
    1. linear OA(32⁹⁸, 1024, F₃₂, 51) (dual of [1024, 926, 52]-code), using an extension C_e(50) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,50], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 51 [i]
    2. linear OA(32⁸², 1024, F₃₂, 43) (dual of [1024, 942, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    3. linear OA(32⁷, 23, F₃₂, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(105−51, 105, 599347)-Net in Base 32 — Upper bound on s

There is no (54, 105, 599348)-net in base 32, because

1 times m-reduction [i] would yield (54, 104, 599348)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 432457 488295 646628 038832 411400 183817 484190 783746 072856 030332 334800 399128 914973 690867 321237 851604 530605 601541 438550 928357 502553 774399 129464 141232 775996 993500 > 32¹⁰⁴ [i]