MinT - Best Known (31, 31+12, s)-Nets in Base 5

Best Known (31, 31+12, s)-Nets in Base 5

(31, 31+12, 208)-Net over F₅ — Constructive and digital

Digital (31, 43, 208)-net over F₅, using

1 times m-reduction [i] based on 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]

(31, 31+12, 722)-Net over F₅ — Digital

Digital (31, 43, 722)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴³, 722, F₅, 12) (dual of [722, 679, 13]-code), using
- 87 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 10 times 0, 1, 23 times 0, 1, 44 times 0) [i] based on linear OA(5³⁷, 629, F₅, 12) (dual of [629, 592, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
    1. linear OA(5³⁷, 625, F₅, 12) (dual of [625, 588, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(5³³, 625, F₅, 11) (dual of [625, 592, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(31, 31+12, 76458)-Net in Base 5 — Upper bound on s

There is no (31, 43, 76459)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 136934 510581 765885 632537 778153 > 5⁴³ [i]