MinT - Best Known (140−33, 140, s)-Nets in Base 5

Best Known (140−33, 140, s)-Nets in Base 5

(140−33, 140, 504)-Net over F₅ — Constructive and digital

Digital (107, 140, 504)-net over F₅, using

5² times duplication [i] based on digital (105, 138, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (36, 52, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 26, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
  2. digital (53, 86, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 43, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(140−33, 140, 3675)-Net over F₅ — Digital

Digital (107, 140, 3675)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁰, 3675, F₅, 33) (dual of [3675, 3535, 34]-code), using
- 536 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 32 times 0, 1, 63 times 0, 1, 103 times 0, 1, 141 times 0, 1, 167 times 0) [i] based on linear OA(5¹³¹, 3130, F₅, 33) (dual of [3130, 2999, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(31) [i] based on
    1. linear OA(5¹³¹, 3125, F₅, 33) (dual of [3125, 2994, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(5¹²⁶, 3125, F₅, 32) (dual of [3125, 2999, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(140−33, 140, 2008066)-Net in Base 5 — Upper bound on s

There is no (107, 140, 2008067)-net in base 5, because

1 times m-reduction [i] would yield (107, 139, 2008067)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 349317 594722 436501 272312 653596 564918 313625 109101 780441 107301 989659 328250 665356 037826 976419 121601 > 5¹³⁹ [i]