MinT - Best Known (200−43, 200, s)-Nets in Base 4

Best Known (200−43, 200, s)-Nets in Base 4

(200−43, 200, 1056)-Net over F₄ — Constructive and digital

Digital (157, 200, 1056)-net over F₄, using

trace code for nets [i] based on digital (7, 50, 264)-net over F₂₅₆, using
- net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(200−43, 200, 4205)-Net over F₄ — Digital

Digital (157, 200, 4205)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁰, 4205, F₄, 43) (dual of [4205, 4005, 44]-code), using
- 89 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 43 times 0) [i] 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]

(200−43, 200, 1467578)-Net in Base 4 — Upper bound on s

There is no (157, 200, 1467579)-net in base 4, because

1 times m-reduction [i] would yield (157, 199, 1467579)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645564 060159 134282 679061 158096 074867 953723 331170 546667 619814 302208 528865 960924 736577 601723 179500 456491 113653 906897 062428 > 4¹⁹⁹ [i]