MinT - Best Known (132, 168, s)-Nets in Base 4

Best Known (132, 168, s)-Nets in Base 4

(132, 168, 1052)-Net over F₄ — Constructive and digital

Digital (132, 168, 1052)-net over F₄, using

trace code for nets [i] based on digital (6, 42, 263)-net over F₂₅₆, using
- net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(132, 168, 4061)-Net over F₄ — Digital

Digital (132, 168, 4061)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁸, 4061, F₄, 36) (dual of [4061, 3893, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁶⁸, 4119, F₄, 36) (dual of [4119, 3951, 37]-code), using
  - strength reduction [i] based on linear OA(4¹⁶⁸, 4119, F₄, 37) (dual of [4119, 3951, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(32) [i] based on
      1. linear OA(4¹⁶³, 4096, F₄, 37) (dual of [4096, 3933, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. 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]
      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, 168, 1047681)-Net in Base 4 — Upper bound on s

There is no (132, 168, 1047682)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 139984 657674 960457 151532 019453 057075 312701 302647 572391 433910 729741 076044 041316 072054 600956 716770 390784 > 4¹⁶⁸ [i]