MinT - Best Known (132−33, 132, s)-Nets in Base 5

Best Known (132−33, 132, s)-Nets in Base 5

(132−33, 132, 408)-Net over F₅ — Constructive and digital

Digital (99, 132, 408)-net over F₅, using

6 times m-reduction [i] based on digital (99, 138, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 69, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(132−33, 132, 2769)-Net over F₅ — Digital

Digital (99, 132, 2769)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³², 2769, F₅, 33) (dual of [2769, 2637, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹³², 3137, F₅, 33) (dual of [3137, 3005, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,15]) [i] based on
    1. linear OA(5¹³¹, 3126, F₅, 33) (dual of [3126, 2995, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(5¹²¹, 3126, F₅, 31) (dual of [3126, 3005, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (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]

(132−33, 132, 898028)-Net in Base 5 — Upper bound on s

There is no (99, 132, 898029)-net in base 5, because

1 times m-reduction [i] would yield (99, 131, 898029)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 36 734576 307843 771170 906054 763306 526929 305946 961956 961859 892618 995524 767981 750606 862030 995905 > 5¹³¹ [i]