MinT - Best Known (113, 113+37, s)-Nets in Base 5

Best Known (113, 113+37, s)-Nets in Base 5

(113, 113+37, 504)-Net over F₅ — Constructive and digital

Digital (113, 150, 504)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (57, 94, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 47, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(113, 113+37, 3164)-Net over F₅ — Digital

Digital (113, 150, 3164)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁵⁰, 3164, F₅, 37) (dual of [3164, 3014, 38]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 7 times 0, 1, 18 times 0) [i] based on linear OA(5¹⁴⁶, 3130, F₅, 37) (dual of [3130, 2984, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(35) [i] based on
    1. linear OA(5¹⁴⁶, 3125, F₅, 37) (dual of [3125, 2979, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(5¹⁴¹, 3125, F₅, 36) (dual of [3125, 2984, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 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]

(113, 113+37, 1153409)-Net in Base 5 — Upper bound on s

There is no (113, 150, 1153410)-net in base 5, because

1 times m-reduction [i] would yield (113, 149, 1153410)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 140 130849 312875 375797 866307 418314 139210 517808 667254 008057 858114 231572 532207 807306 798648 896601 036965 239105 > 5¹⁴⁹ [i]