MinT - Best Known (72, 72+22, s)-Nets in Base 4

Best Known (72, 72+22, s)-Nets in Base 4

(72, 72+22, 1032)-Net over F₄ — Constructive and digital

Digital (72, 94, 1032)-net over F₄, using

4² times duplication [i] based on digital (70, 92, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 23, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(72, 72+22, 1445)-Net over F₄ — Digital

Digital (72, 94, 1445)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁴, 1445, F₄, 22) (dual of [1445, 1351, 23]-code), using
- 403 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 17 times 0, 1, 28 times 0, 1, 42 times 0, 1, 55 times 0, 1, 68 times 0, 1, 78 times 0, 1, 87 times 0) [i] based on linear OA(4⁸¹, 1029, F₄, 22) (dual of [1029, 948, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. linear OA(4⁸¹, 1024, F₄, 22) (dual of [1024, 943, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(4⁷⁶, 1024, F₄, 21) (dual of [1024, 948, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(72, 72+22, 228429)-Net in Base 4 — Upper bound on s

There is no (72, 94, 228430)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 392 323085 639624 079754 565203 365179 723752 864959 845472 379426 > 4⁹⁴ [i]