MinT - Best Known (48−16, 48, s)-Nets in Base 5

Best Known (48−16, 48, s)-Nets in Base 5

(48−16, 48, 138)-Net over F₅ — Constructive and digital

Digital (32, 48, 138)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (24, 40, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 20, 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]

(48−16, 48, 318)-Net over F₅ — Digital

Digital (32, 48, 318)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴⁸, 318, F₅, 16) (dual of [318, 270, 17]-code), using
- constructionÂ X applied to C_e(15) âŠ‚ C_e(13) [i] based on
  1. 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]
  2. 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]
  3. linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(48−16, 48, 14698)-Net in Base 5 — Upper bound on s

There is no (32, 48, 14699)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3552 722757 910677 045268 397947 549409 > 5⁴⁸ [i]