MinT - Best Known (33, 33+9, s)-Nets in Base 4

Best Known (33, 33+9, s)-Nets in Base 4

(33, 33+9, 1365)-Net over F₄ — Constructive and digital

Digital (33, 42, 1365)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁴², 1365, F₄, 9, 9) (dual of [(1365, 9), 12243, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4⁴², 5461, F₄, 9) (dual of [5461, 5419, 10]-code), using
  - the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,820,1639,…,1092}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(33, 33+9, 3781)-Net over F₄ — Digital

Digital (33, 42, 3781)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴², 3781, F₄, 9) (dual of [3781, 3739, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴², 5461, F₄, 9) (dual of [5461, 5419, 10]-code), using
  - the BCH-code C(I) with length 5461Â | 4⁷âˆ’1, defining interval IÂ =Â {1,820,1639,…,1092}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(33, 33+9, 1094070)-Net in Base 4 — Upper bound on s

There is no (33, 42, 1094071)-net in base 4, because

1 times m-reduction [i] would yield (33, 41, 1094071)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4 835710 602079 155814 298251 > 4⁴¹ [i]