MinT - Best Known (89, 89+27, s)-Nets in Base 5

Best Known (89, 89+27, s)-Nets in Base 5

(89, 89+27, 416)-Net over F₅ — Constructive and digital

Digital (89, 116, 416)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (31, 44, 208)-net over F₅, using
  - trace code for nets [i] based on digital (9, 22, 104)-net over F₂₅, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
        a function field by GarcÃa and Quoos [i]
2. digital (45, 72, 208)-net over F₅, using
  - trace code for nets [i] based on digital (9, 36, 104)-net over F₂₅, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₂₅ (see above)

(89, 89+27, 3532)-Net over F₅ — Digital

Digital (89, 116, 3532)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹⁶, 3532, F₅, 27) (dual of [3532, 3416, 28]-code), using
- 392 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 37 times 0, 1, 66 times 0, 1, 104 times 0, 1, 147 times 0) [i] based on linear OA(5¹⁰⁶, 3130, F₅, 27) (dual of [3130, 3024, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(25) [i] based on
    1. linear OA(5¹⁰⁶, 3125, F₅, 27) (dual of [3125, 3019, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. linear OA(5¹⁰¹, 3125, F₅, 26) (dual of [3125, 3024, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [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]

(89, 89+27, 2160431)-Net in Base 5 — Upper bound on s

There is no (89, 116, 2160432)-net in base 5, because

1 times m-reduction [i] would yield (89, 115, 2160432)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 240 741832 164737 995694 846495 631136 013209 644816 169410 165199 410920 633596 327957 612225 > 5¹¹⁵ [i]