MinT - Best Known (159−35, 159, s)-Nets in Base 4

Best Known (159−35, 159, s)-Nets in Base 4

(159−35, 159, 1044)-Net over F₄ — Constructive and digital

Digital (124, 159, 1044)-net over F₄, using

1 times m-reduction [i] based on digital (124, 160, 1044)-net over F₄, using
- trace code for nets [i] based on digital (4, 40, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(159−35, 159, 3322)-Net over F₄ — Digital

Digital (124, 159, 3322)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁹, 3322, F₄, 35) (dual of [3322, 3163, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁵⁹, 4111, F₄, 35) (dual of [4111, 3952, 36]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁵⁸, 4110, F₄, 35) (dual of [4110, 3952, 36]-code), using
    - constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(4¹⁵⁷, 4097, F₄, 35) (dual of [4097, 3940, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
      2. linear OA(4¹⁴⁵, 4097, F₄, 33) (dual of [4097, 3952, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(159−35, 159, 942831)-Net in Base 4 — Upper bound on s

There is no (124, 159, 942832)-net in base 4, because

1 times m-reduction [i] would yield (124, 158, 942832)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 133500 586613 721823 412686 976453 087294 693860 427695 756290 047979 022896 962105 245705 538827 552225 036630 > 4¹⁵⁸ [i]