MinT - Best Known (241−33, 241, s)-Nets in Base 4

Best Known (241−33, 241, s)-Nets in Base 4

Digital (208, 241, 65536)-net over F₄, using

net defined by OOA [i] based on linear OOA(4²⁴¹, 65536, F₄, 33, 33) (dual of [(65536, 33), 2162447, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4²⁴¹, 1048577, F₄, 33) (dual of [1048577, 1048336, 34]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

Digital (208, 241, 339637)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁴¹, 339637, F₄, 3, 33) (dual of [(339637, 3), 1018670, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4²⁴¹, 349525, F₄, 3, 33) (dual of [(349525, 3), 1048334, 34]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(4²⁴¹, 1048575, F₄, 33) (dual of [1048575, 1048334, 34]-code), using
    - discarding factors / shortening the dual code based on linear OA(4²⁴¹, 1048576, F₄, 33) (dual of [1048576, 1048335, 34]-code), using
      - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

There is no (208, 241, large)-net in base 4, because

31 times m-reduction [i] would yield (208, 210, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]