MinT - Best Known (77, 77+25, s)-Nets in Base 5

Best Known (77, 77+25, s)-Nets in Base 5

(77, 77+25, 400)-Net over F₅ — Constructive and digital

Digital (77, 102, 400)-net over F₅, using

2 times m-reduction [i] based on digital (77, 104, 400)-net over F₅, using
- trace code for nets [i] based on digital (25, 52, 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]

(77, 77+25, 2750)-Net over F₅ — Digital

Digital (77, 102, 2750)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰², 2750, F₅, 25) (dual of [2750, 2648, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁰², 3137, F₅, 25) (dual of [3137, 3035, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,11]) [i] based on
    1. linear OA(5¹⁰¹, 3126, F₅, 25) (dual of [3126, 3025, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. 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]
    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]

(77, 77+25, 1009939)-Net in Base 5 — Upper bound on s

There is no (77, 102, 1009940)-net in base 5, because

1 times m-reduction [i] would yield (77, 101, 1009940)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 39443 339138 101135 376953 469958 274502 511790 706448 439298 462203 126517 185281 > 5¹⁰¹ [i]