MinT - Best Known (140, 140+32, s)-Nets in Base 4

Best Known (140, 140+32, s)-Nets in Base 4

(140, 140+32, 1158)-Net over F₄ — Constructive and digital

Digital (140, 172, 1158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (28, 44, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 22, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
2. digital (96, 128, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 32, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(140, 140+32, 10826)-Net over F₄ — Digital

Digital (140, 172, 10826)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷², 10826, F₄, 32) (dual of [10826, 10654, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷², 16402, F₄, 32) (dual of [16402, 16230, 33]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(4¹⁶⁹, 16385, F₄, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹⁵⁵, 16385, F₄, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(4³, 17, F₄, 2) (dual of [17, 14, 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]

(140, 140+32, 6722975)-Net in Base 4 — Upper bound on s

There is no (140, 172, 6722976)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 835933 352900 118495 818758 953425 114164 343606 723646 878201 264354 593566 041827 548219 188819 068060 252651 436891 > 4¹⁷² [i]