MinT - Best Known (121, 121+33, s)-Nets in Base 4

Best Known (121, 121+33, s)-Nets in Base 4

(121, 121+33, 1048)-Net over F₄ — Constructive and digital

Digital (121, 154, 1048)-net over F₄, using

4² times duplication [i] based on digital (119, 152, 1048)-net over F₄, using
- trace code for nets [i] based on digital (5, 38, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(121, 121+33, 3851)-Net over F₄ — Digital

Digital (121, 154, 3851)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁴, 3851, F₄, 33) (dual of [3851, 3697, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁵⁴, 4121, F₄, 33) (dual of [4121, 3967, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,13]) [i] based on
    1. 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]
    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(4⁹, 24, F₄, 5) (dual of [24, 15, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁹, 36, F₄, 5) (dual of [36, 27, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(121, 121+33, 1296020)-Net in Base 4 — Upper bound on s

There is no (121, 154, 1296021)-net in base 4, because

1 times m-reduction [i] would yield (121, 153, 1296021)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 371273 094289 026686 639282 344053 070644 343197 765425 665626 106736 407140 997580 021289 180085 189509 > 4¹⁵³ [i]