MinT - Best Known (150−33, 150, s)-Nets in Base 4

Best Known (150−33, 150, s)-Nets in Base 4

(150−33, 150, 1044)-Net over F₄ — Constructive and digital

Digital (117, 150, 1044)-net over F₄, using

4² times duplication [i] based on digital (115, 148, 1044)-net over F₄, using
- trace code for nets [i] based on digital (4, 37, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(150−33, 150, 3217)-Net over F₄ — Digital

Digital (117, 150, 3217)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁰, 3217, F₄, 33) (dual of [3217, 3067, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁵⁰, 4119, F₄, 33) (dual of [4119, 3969, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(28) [i] based on
    1. linear OA(4¹⁴⁵, 4096, F₄, 33) (dual of [4096, 3951, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(4¹²⁷, 4096, F₄, 29) (dual of [4096, 3969, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(4⁵, 23, F₄, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(150−33, 150, 916421)-Net in Base 4 — Upper bound on s

There is no (117, 150, 916422)-net in base 4, because

1 times m-reduction [i] would yield (117, 149, 916422)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 509267 418761 862691 453149 504544 480049 435338 907675 194792 699488 085566 939061 369867 936503 206932 > 4¹⁴⁹ [i]