MinT - Best Known (51−17, 51, s)-Nets in Base 5

Best Known (51−17, 51, s)-Nets in Base 5

(51−17, 51, 142)-Net over F₅ — Constructive and digital

Digital (34, 51, 142)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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 (25, 42, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 21, 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]

(51−17, 51, 317)-Net over F₅ — Digital

Digital (34, 51, 317)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵¹, 317, F₅, 17) (dual of [317, 266, 18]-code), using
- constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
  1. linear OA(5⁵¹, 313, F₅, 17) (dual of [313, 262, 18]-code), using an extension C_e(16) of the narrow-sense BCH-code C(I) with length 312Â | 5⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
  2. linear OA(5⁴⁷, 313, F₅, 16) (dual of [313, 266, 17]-code), using an extension C_e(15) of the narrow-sense BCH-code C(I) with length 312Â | 5⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
  3. linear OA(5⁰, 4, F₅, 0) (dual of [4, 4, 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]

(51−17, 51, 21982)-Net in Base 5 — Upper bound on s

There is no (34, 51, 21983)-net in base 5, because

1 times m-reduction [i] would yield (34, 50, 21983)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 88822 738055 349200 290955 063068 706785 > 5⁵⁰ [i]