MinT - Best Known (155, 198, s)-Nets in Base 4

Best Known (155, 198, s)-Nets in Base 4

(155, 198, 1052)-Net over F₄ — Constructive and digital

Digital (155, 198, 1052)-net over F₄, using

4² times duplication [i] based on digital (153, 196, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(155, 198, 4135)-Net over F₄ — Digital

Digital (155, 198, 4135)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁸, 4135, F₄, 43) (dual of [4135, 3937, 44]-code), using
- 21 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 5 times 0, 1, 12 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]

(155, 198, 1286061)-Net in Base 4 — Upper bound on s

There is no (155, 198, 1286062)-net in base 4, because

1 times m-reduction [i] would yield (155, 197, 1286062)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40348 093649 715411 569561 398000 555663 301902 156491 980470 699203 362772 416373 375530 956458 483104 810805 665434 531596 312569 668162 > 4¹⁹⁷ [i]