MinT - Best Known (34−12, 34, s)-Nets in Base 5

Best Known (34−12, 34, s)-Nets in Base 5

(34−12, 34, 132)-Net over F₅ — Constructive and digital

Digital (22, 34, 132)-net over F₅, using

2 times m-reduction [i] based on digital (22, 36, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(34−12, 34, 185)-Net over F₅ — Digital

Digital (22, 34, 185)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³⁴, 185, F₅, 12) (dual of [185, 151, 13]-code), using
- 51 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 14 times 0, 1, 20 times 0) [i] based on linear OA(5²⁸, 128, F₅, 12) (dual of [128, 100, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
    1. linear OA(5²⁸, 125, F₅, 12) (dual of [125, 97, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(5²⁵, 125, F₅, 11) (dual of [125, 100, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(5⁰, 3, F₅, 0) (dual of [3, 3, 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]

(34−12, 34, 6834)-Net in Base 5 — Upper bound on s

There is no (22, 34, 6835)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 582080 072683 377441 793769 > 5³⁴ [i]