MinT - Best Known (76, 76+23, s)-Nets in Base 5

Best Known (76, 76+23, s)-Nets in Base 5

(76, 76+23, 400)-Net over F₅ — Constructive and digital

Digital (76, 99, 400)-net over F₅, using

3 times m-reduction [i] based on digital (76, 102, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 51, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

(76, 76+23, 3325)-Net over F₅ — Digital

Digital (76, 99, 3325)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁹, 3325, F₅, 23) (dual of [3325, 3226, 24]-code), using
- 181 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 48 times 0, 1, 92 times 0) [i] based on linear OA(5⁹², 3137, F₅, 23) (dual of [3137, 3045, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(5⁹¹, 3126, F₅, 23) (dual of [3126, 3035, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(5⁸¹, 3126, F₅, 21) (dual of [3126, 3045, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 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+23, 2070808)-Net in Base 5 — Upper bound on s

There is no (76, 99, 2070809)-net in base 5, because

1 times m-reduction [i] would yield (76, 98, 2070809)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 315 545904 116540 743131 408474 630743 381997 813382 422451 286306 239741 975485 > 5⁹⁸ [i]