MinT - Best Known (72, 94, s)-Nets in Base 5

Best Known (72, 94, s)-Nets in Base 5

(72, 94, 400)-Net over F₅ — Constructive and digital

Digital (72, 94, 400)-net over F₅, using

trace code for nets [i] based on digital (25, 47, 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]

(72, 94, 3271)-Net over F₅ — Digital

Digital (72, 94, 3271)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁴, 3271, F₅, 22) (dual of [3271, 3177, 23]-code), using
- 133 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0) [i] based on linear OA(5⁸⁶, 3130, F₅, 22) (dual of [3130, 3044, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. linear OA(5⁸⁶, 3125, F₅, 22) (dual of [3125, 3039, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(5⁸¹, 3125, F₅, 21) (dual of [3125, 3044, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(72, 94, 1153367)-Net in Base 5 — Upper bound on s

There is no (72, 94, 1153368)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 504874 182285 354280 146108 731580 957900 548079 149938 964763 434106 844769 > 5⁹⁴ [i]