MinT - Best Known (89, 89+25, s)-Nets in Base 4

Best Known (89, 89+25, s)-Nets in Base 4

(89, 89+25, 1040)-Net over F₄ — Constructive and digital

Digital (89, 114, 1040)-net over F₄, using

4² times duplication [i] based on digital (87, 112, 1040)-net over F₄, using
- trace code for nets [i] based on digital (3, 28, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(89, 89+25, 2835)-Net over F₄ — Digital

Digital (89, 114, 2835)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁴, 2835, F₄, 25) (dual of [2835, 2721, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁴, 4119, F₄, 25) (dual of [4119, 4005, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(20) [i] based on
    1. linear OA(4¹⁰⁹, 4096, F₄, 25) (dual of [4096, 3987, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(4⁹¹, 4096, F₄, 21) (dual of [4096, 4005, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(89, 89+25, 823442)-Net in Base 4 — Upper bound on s

There is no (89, 114, 823443)-net in base 4, because

1 times m-reduction [i] would yield (89, 113, 823443)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 107 840209 060695 586810 714203 262461 500257 520723 230810 598711 156267 522020 > 4¹¹³ [i]