MinT - Best Known (48, 48+46, s)-Nets in Base 32

Best Known (48, 48+46, s)-Nets in Base 32

(48, 48+46, 240)-Net over F₃₂ — Constructive and digital

Digital (48, 94, 240)-net over F₃₂, using

6 times m-reduction [i] based on digital (48, 100, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 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, 63, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(48, 48+46, 513)-Net in Base 32 — Constructive

(48, 94, 513)-net in base 32, using

t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 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]

(48, 48+46, 825)-Net over F₃₂ — Digital

Digital (48, 94, 825)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁴, 825, F₃₂, 46) (dual of [825, 731, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁴, 1044, F₃₂, 46) (dual of [1044, 950, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(38) [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(32⁷⁴, 1024, F₃₂, 39) (dual of [1024, 950, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    3. linear OA(32⁶, 20, F₃₂, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁶, 32, F₃₂, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
        Reedâ€“Solomon code RS(26,32) [i]

(48, 48+46, 431087)-Net in Base 32 — Upper bound on s

There is no (48, 94, 431088)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3048 688652 439764 923791 753357 181901 164660 526325 718313 197746 683755 413124 860023 412757 232781 039209 218821 826202 296228 364519 437424 229719 735134 722942 > 32⁹⁴ [i]