MinT - Best Known (194−43, 194, s)-Nets in Base 4

Best Known (194−43, 194, s)-Nets in Base 4

(194−43, 194, 1048)-Net over F₄ — Constructive and digital

Digital (151, 194, 1048)-net over F₄, using

4² times duplication [i] based on digital (149, 192, 1048)-net over F₄, using
- trace code for nets [i] based on digital (5, 48, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(194−43, 194, 3639)-Net over F₄ — Digital

Digital (151, 194, 3639)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁴, 3639, F₄, 43) (dual of [3639, 3445, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁴, 4110, F₄, 43) (dual of [4110, 3916, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
    1. linear OA(4¹⁹³, 4097, F₄, 43) (dual of [4097, 3904, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(4¹⁸¹, 4097, F₄, 41) (dual of [4097, 3916, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (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]

(194−43, 194, 987601)-Net in Base 4 — Upper bound on s

There is no (151, 194, 987602)-net in base 4, because

1 times m-reduction [i] would yield (151, 193, 987602)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 608850 080262 052850 278814 151832 521805 599942 632724 807282 263588 232761 207534 787476 266201 602982 576183 992427 414302 010032 > 4¹⁹³ [i]