MinT - Best Known (24, 31, s)-Nets in Base 4

Best Known (24, 31, s)-Nets in Base 4

(24, 31, 1367)-Net over F₄ — Constructive and digital

Digital (24, 31, 1367)-net over F₄, using

net defined by OOA [i] based on linear OOA(4³¹, 1367, F₄, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(4³¹, 4102, F₄, 7) (dual of [4102, 4071, 8]-code), using
  - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
    1. linear OA(4³¹, 4096, F₄, 7) (dual of [4096, 4065, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. linear OA(4²⁵, 4096, F₄, 6) (dual of [4096, 4071, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

(24, 31, 3554)-Net over F₄ — Digital

Digital (24, 31, 3554)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³¹, 3554, F₄, 7) (dual of [3554, 3523, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4³¹, 4096, F₄, 7) (dual of [4096, 4065, 8]-code), using
  - an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(24, 31, 635127)-Net in Base 4 — Upper bound on s

There is no (24, 31, 635128)-net in base 4, because

1 times m-reduction [i] would yield (24, 30, 635128)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 152923 242245 846085 > 4³⁰ [i]