MinT - Best Known (38, 38+16, s)-Nets in Base 5

Best Known (38, 38+16, s)-Nets in Base 5

(38, 38+16, 252)-Net over F₅ — Constructive and digital

Digital (38, 54, 252)-net over F₅, using

2 times m-reduction [i] based on digital (38, 56, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 28, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(38, 38+16, 647)-Net over F₅ — Digital

Digital (38, 54, 647)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁴, 647, F₅, 16) (dual of [647, 593, 17]-code), using
- 13 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 0, 1, 7 times 0) [i] based on linear OA(5⁵⁰, 630, F₅, 16) (dual of [630, 580, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(13) [i] based on
    1. linear OA(5⁴⁹, 625, F₅, 16) (dual of [625, 576, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(5⁴⁵, 625, F₅, 14) (dual of [625, 580, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(38, 38+16, 49161)-Net in Base 5 — Upper bound on s

There is no (38, 54, 49162)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 55 511684 027269 261424 941279 354840 968065 > 5⁵⁴ [i]