Best Known (40−5, 40, s)-Nets in Base 4
(40−5, 40, 4194322)-Net over F4 — Constructive and digital
Digital (35, 40, 4194322)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 3, 21)-net over F4, using
- digital (32, 37, 4194301)-net over F4, using
- net defined by OOA [i] based on linear OOA(437, 4194301, F4, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
- net defined by OOA [i] based on linear OOA(437, 4194301, F4, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
(40−5, 40, large)-Net over F4 — Digital
Digital (35, 40, large)-net over F4, using
- 43 times duplication [i] based on digital (32, 37, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(437, large, F4, 5) (dual of [large, large−37, 6]-code), using
(40−5, 40, large)-Net in Base 4 — Upper bound on s
There is no (35, 40, large)-net in base 4, because
- 3 times m-reduction [i] would yield (35, 37, large)-net in base 4, but