MinT - Best Known (132−28, 132, s)-Nets in Base 4

Best Known (132−28, 132, s)-Nets in Base 4

(132−28, 132, 1048)-Net over F₄ — Constructive and digital

Digital (104, 132, 1048)-net over F₄, using

trace code for nets [i] based on digital (5, 33, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(132−28, 132, 3778)-Net over F₄ — Digital

Digital (104, 132, 3778)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³², 3778, F₄, 28) (dual of [3778, 3646, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³², 4119, F₄, 28) (dual of [4119, 3987, 29]-code), using
  - strength reduction [i] based on linear OA(4¹³², 4119, F₄, 29) (dual of [4119, 3987, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(24) [i] based on
      1. 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]
      2. 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]
      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]

(132−28, 132, 956967)-Net in Base 4 — Upper bound on s

There is no (104, 132, 956968)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 29 643124 660141 621892 913011 547093 031956 344195 635553 504995 536080 485120 465632 946060 > 4¹³² [i]