MinT - Best Known (92, 92+26, s)-Nets in Base 4

Best Known (92, 92+26, s)-Nets in Base 4

(92, 92+26, 1040)-Net over F₄ — Constructive and digital

Digital (92, 118, 1040)-net over F₄, using

4² times duplication [i] based on digital (90, 116, 1040)-net over F₄, using
- trace code for nets [i] based on digital (3, 29, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(92, 92+26, 2795)-Net over F₄ — Digital

Digital (92, 118, 2795)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁸, 2795, F₄, 26) (dual of [2795, 2677, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁸, 4111, F₄, 26) (dual of [4111, 3993, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(22) [i] based on
    1. linear OA(4¹¹⁵, 4096, F₄, 26) (dual of [4096, 3981, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(4¹⁰³, 4096, F₄, 23) (dual of [4096, 3993, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(4³, 15, F₄, 2) (dual of [15, 12, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(92, 92+26, 550971)-Net in Base 4 — Upper bound on s

There is no (92, 118, 550972)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 110429 664988 582725 010108 010461 556189 036178 536796 853911 565400 044951 854748 > 4¹¹⁸ [i]