MinT - Best Known (72, 72+32, s)-Nets in Base 5

Best Known (72, 72+32, s)-Nets in Base 5

(72, 72+32, 296)-Net over F₅ — Constructive and digital

Digital (72, 104, 296)-net over F₅, using

2 times m-reduction [i] based on digital (72, 106, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 53, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

(72, 72+32, 704)-Net over F₅ — Digital

Digital (72, 104, 704)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁴, 704, F₅, 32) (dual of [704, 600, 33]-code), using
- 70 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 11 times 0, 1, 22 times 0, 1, 30 times 0) [i] based on linear OA(5⁹⁹, 629, F₅, 32) (dual of [629, 530, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(30) [i] based on
    1. linear OA(5⁹⁹, 625, F₅, 32) (dual of [625, 526, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. linear OA(5⁹⁵, 625, F₅, 31) (dual of [625, 530, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [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]

(72, 72+32, 59388)-Net in Base 5 — Upper bound on s

There is no (72, 104, 59389)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4 931531 176183 516511 459929 282124 325400 426652 724910 237458 919665 417298 610625 > 5¹⁰⁴ [i]