MinT - Best Known (141−33, 141, s)-Nets in Base 5

Best Known (141−33, 141, s)-Nets in Base 5

(141−33, 141, 504)-Net over F₅ — Constructive and digital

Digital (108, 141, 504)-net over F₅, using

1 times m-reduction [i] based on digital (108, 142, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (37, 54, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 27, 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 (54, 88, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 44, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(141−33, 141, 3860)-Net over F₅ — Digital

Digital (108, 141, 3860)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴¹, 3860, F₅, 33) (dual of [3860, 3719, 34]-code), using
- 720 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, 1, 183 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]

(141−33, 141, 2220567)-Net in Base 5 — Upper bound on s

There is no (108, 141, 2220568)-net in base 5, because

1 times m-reduction [i] would yield (108, 140, 2220568)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 71 746626 591016 456865 775634 385005 797360 504360 821065 641991 722532 111995 070686 479499 351753 540610 308097 > 5¹⁴⁰ [i]