MinT - Best Known (44, 44+39, s)-Nets in Base 32

Best Known (44, 44+39, s)-Nets in Base 32

(44, 44+39, 240)-Net over F₃₂ — Constructive and digital

Digital (44, 83, 240)-net over F₃₂, using

5 times m-reduction [i] based on digital (44, 88, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(44, 44+39, 513)-Net in Base 32 — Constructive

(44, 83, 513)-net in base 32, using

13 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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]

(44, 44+39, 1007)-Net over F₃₂ — Digital

Digital (44, 83, 1007)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸³, 1007, F₃₂, 39) (dual of [1007, 924, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸³, 1050, F₃₂, 39) (dual of [1050, 967, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(28) [i] based on
    1. 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]
    2. linear OA(32⁵⁷, 1024, F₃₂, 29) (dual of [1024, 967, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(32⁹, 26, F₃₂, 9) (dual of [26, 17, 10]-code or 26-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]

(44, 44+39, 801233)-Net in Base 32 — Upper bound on s

There is no (44, 83, 801234)-net in base 32, because

1 times m-reduction [i] would yield (44, 82, 801234)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2644 262659 469358 444290 279916 533432 231998 376786 237612 588731 684487 347401 662724 421299 378275 017344 392749 656101 139579 550472 039200 > 32⁸² [i]