MinT - Best Known (85, 85+27, s)-Nets in Base 5

Best Known (85, 85+27, s)-Nets in Base 5

(85, 85+27, 400)-Net over F₅ — Constructive and digital

Digital (85, 112, 400)-net over F₅, using

8 times m-reduction [i] based on digital (85, 120, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 60, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

(85, 85+27, 3170)-Net over F₅ — Digital

Digital (85, 112, 3170)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹², 3170, F₅, 27) (dual of [3170, 3058, 28]-code), using
- 34 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 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]

(85, 85+27, 1316653)-Net in Base 5 — Upper bound on s

There is no (85, 112, 1316654)-net in base 5, because

1 times m-reduction [i] would yield (85, 111, 1316654)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 385186 317638 947294 329378 042602 938570 782468 742471 941846 860960 620719 099057 700665 > 5¹¹¹ [i]