MinT - Best Known (134−29, 134, s)-Nets in Base 4

Best Known (134−29, 134, s)-Nets in Base 4

(134−29, 134, 1044)-Net over F₄ — Constructive and digital

Digital (105, 134, 1044)-net over F₄, using

4² times duplication [i] based on digital (103, 132, 1044)-net over F₄, using
- trace code for nets [i] based on digital (4, 33, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(134−29, 134, 3344)-Net over F₄ — Digital

Digital (105, 134, 3344)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁴, 3344, F₄, 29) (dual of [3344, 3210, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³⁴, 4110, F₄, 29) (dual of [4110, 3976, 30]-code), using
  - constructionÂ X applied to C([0,14]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(4¹³³, 4097, F₄, 29) (dual of [4097, 3964, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    2. linear OA(4¹²¹, 4097, F₄, 27) (dual of [4097, 3976, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (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]

(134−29, 134, 1056578)-Net in Base 4 — Upper bound on s

There is no (105, 134, 1056579)-net in base 4, because

1 times m-reduction [i] would yield (105, 133, 1056579)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 118 571719 305314 642239 281208 554463 518975 056613 396735 353716 426574 762403 463418 166064 > 4¹³³ [i]