MinT - Best Known (116−26, 116, s)-Nets in Base 4

Best Known (116−26, 116, s)-Nets in Base 4

(116−26, 116, 1040)-Net over F₄ — Constructive and digital

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]

(116−26, 116, 2488)-Net over F₄ — Digital

Digital (90, 116, 2488)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁶, 2488, F₄, 26) (dual of [2488, 2372, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁶, 4103, F₄, 26) (dual of [4103, 3987, 27]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹¹⁵, 4102, F₄, 26) (dual of [4102, 3987, 27]-code), using
    - constructionÂ X applied to C_e(25) âŠ‚ C_e(24) [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(4¹⁰⁹, 4096, F₄, 25) (dual of [4096, 3987, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(116−26, 116, 445145)-Net in Base 4 — Upper bound on s

There is no (90, 116, 445146)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6901 750080 711946 696170 100760 072893 951034 715174 076320 809498 000529 027844 > 4¹¹⁶ [i]