MinT - Best Known (30, 30+14, s)-Nets in Base 5

Best Known (30, 30+14, s)-Nets in Base 5

(30, 30+14, 142)-Net over F₅ — Constructive and digital

Digital (30, 44, 142)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (22, 36, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 18, 66)-net over F₂₅, using
    - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
        a function field by GarcÃa and Quoos [i]

(30, 30+14, 341)-Net over F₅ — Digital

Digital (30, 44, 341)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴⁴, 341, F₅, 14) (dual of [341, 297, 15]-code), using
- 25 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 24 times 0) [i] based on linear OA(5⁴³, 315, F₅, 14) (dual of [315, 272, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. linear OA(5⁴³, 313, F₅, 14) (dual of [313, 270, 15]-code), using an extension C_e(13) of the narrow-sense BCH-code C(I) with length 312Â | 5⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(5⁴¹, 313, F₅, 13) (dual of [313, 272, 14]-code), using an extension C_e(12) of the narrow-sense BCH-code C(I) with length 312Â | 5⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(5⁰, 2, F₅, 0) (dual of [2, 2, 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]

(30, 30+14, 20906)-Net in Base 5 — Upper bound on s

There is no (30, 44, 20907)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5 684889 025556 136738 684920 073685 > 5⁴⁴ [i]