MinT - Best Known (76, 76+36, s)-Nets in Base 5

Best Known (76, 76+36, s)-Nets in Base 5

(76, 76+36, 296)-Net over F₅ — Constructive and digital

Digital (76, 112, 296)-net over F₅, using

2 times m-reduction [i] based on digital (76, 114, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 57, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

(76, 76+36, 626)-Net over F₅ — Digital

Digital (76, 112, 626)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹², 626, F₅, 36) (dual of [626, 514, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹¹², 630, F₅, 36) (dual of [630, 518, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(33) [i] based on
    1. linear OA(5¹¹¹, 625, F₅, 36) (dual of [625, 514, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    2. linear OA(5¹⁰⁷, 625, F₅, 34) (dual of [625, 518, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(76, 76+36, 42177)-Net in Base 5 — Upper bound on s

There is no (76, 112, 42178)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 926173 028439 931739 238733 664043 595731 445668 449215 833417 630804 463509 496259 196225 > 5¹¹² [i]