MinT - Best Known (32, 32+12, s)-Nets in Base 5

Best Known (32, 32+12, s)-Nets in Base 5

(32, 32+12, 252)-Net over F₅ — Constructive and digital

Digital (32, 44, 252)-net over F₅, using

trace code for nets [i] based on digital (10, 22, 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]

(32, 32+12, 797)-Net over F₅ — Digital

Digital (32, 44, 797)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴⁴, 797, F₅, 12) (dual of [797, 753, 13]-code), using
- 161 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, 1, 73 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]

(32, 32+12, 99982)-Net in Base 5 — Upper bound on s

There is no (32, 44, 99983)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5 684452 976472 391907 735554 414057 > 5⁴⁴ [i]