MinT - Best Known (130, 164, s)-Nets in Base 4

Best Known (130, 164, s)-Nets in Base 4

(130, 164, 1056)-Net over F₄ — Constructive and digital

Digital (130, 164, 1056)-net over F₄, using

trace code for nets [i] based on digital (7, 41, 264)-net over F₂₅₆, using
- net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(130, 164, 4430)-Net over F₄ — Digital

Digital (130, 164, 4430)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁴, 4430, F₄, 34) (dual of [4430, 4266, 35]-code), using
- 315 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 20 times 0, 1, 33 times 0, 1, 50 times 0, 1, 73 times 0, 1, 101 times 0) [i] based on linear OA(4¹⁵¹, 4102, F₄, 34) (dual of [4102, 3951, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
    1. linear OA(4¹⁵¹, 4096, F₄, 34) (dual of [4096, 3945, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(130, 164, 1537900)-Net in Base 4 — Upper bound on s

There is no (130, 164, 1537901)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 815570 771565 544915 079605 348000 668293 538811 416918 512141 575293 685443 132148 030741 401048 526918 759504 > 4¹⁶⁴ [i]