MinT - Best Known (140, 140+18, s)-Nets in Base 4

Best Known (140, 140+18, s)-Nets in Base 4

(140, 140+18, 932067)-Net over F₄ — Constructive and digital

Digital (140, 158, 932067)-net over F₄, using

4¹ times duplication [i] based on digital (139, 157, 932067)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹⁵⁷, 932067, F₄, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
  - OA 9-folding and stacking [i] based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(140, 140+18, 3909021)-Net over F₄ — Digital

Digital (140, 158, 3909021)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁵⁸, 3909021, F₄, 2, 18) (dual of [(3909021, 2), 7817884, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4¹⁵⁸, 4194301, F₄, 2, 18) (dual of [(4194301, 2), 8388444, 19]-NRT-code), using
  - 4¹ times duplication [i] based on linear OOA(4¹⁵⁷, 4194301, F₄, 2, 18) (dual of [(4194301, 2), 8388445, 19]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4¹⁵⁷, 8388602, F₄, 18) (dual of [8388602, 8388445, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(140, 140+18, large)-Net in Base 4 — Upper bound on s

There is no (140, 158, large)-net in base 4, because

16 times m-reduction [i] would yield (140, 142, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]