MinT - Best Known (78−18, 78, s)-Nets in Base 4

Best Known (78−18, 78, s)-Nets in Base 4

(78−18, 78, 1032)-Net over F₄ — Constructive and digital

Digital (60, 78, 1032)-net over F₄, using

4² times duplication [i] based on digital (58, 76, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 19, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(78−18, 78, 1401)-Net over F₄ — Digital

Digital (60, 78, 1401)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷⁸, 1401, F₄, 18) (dual of [1401, 1323, 19]-code), using
- 360 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 18 times 0, 1, 30 times 0, 1, 45 times 0, 1, 63 times 0, 1, 80 times 0, 1, 97 times 0) [i] based on linear OA(4⁶⁶, 1029, F₄, 18) (dual of [1029, 963, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(4⁶⁶, 1024, F₄, 18) (dual of [1024, 958, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(4⁶¹, 1024, F₄, 17) (dual of [1024, 963, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(4⁰, 5, F₄, 0) (dual of [5, 5, 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]

(78−18, 78, 228281)-Net in Base 4 — Upper bound on s

There is no (60, 78, 228282)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 91346 756155 436010 392715 781471 735914 877980 047773 > 4⁷⁸ [i]