MinT - Best Known (180−38, 180, s)-Nets in Base 4

Best Known (180−38, 180, s)-Nets in Base 4

(180−38, 180, 1056)-Net over F₄ — Constructive and digital

Digital (142, 180, 1056)-net over F₄, using

trace code for nets [i] based on digital (7, 45, 264)-net over F₂₅₆, using
- net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(180−38, 180, 4311)-Net over F₄ — Digital

Digital (142, 180, 4311)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁰, 4311, F₄, 38) (dual of [4311, 4131, 39]-code), using
- 198 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0, 1, 47 times 0, 1, 68 times 0) [i] based on linear OA(4¹⁶⁹, 4102, F₄, 38) (dual of [4102, 3933, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
    1. linear OA(4¹⁶⁹, 4096, F₄, 38) (dual of [4096, 3927, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. 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]
    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]

(180−38, 180, 1336027)-Net in Base 4 — Upper bound on s

There is no (142, 180, 1336028)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 348548 644164 388676 650657 086578 661687 912858 708960 411475 663460 147730 792017 338629 130667 882491 023455 771235 428969 > 4¹⁸⁰ [i]