Best Known (178, 178+21, s)-Nets in Base 4
(178, 178+21, 838881)-Net over F4 — Constructive and digital
Digital (178, 199, 838881)-net over F4, using
- 41 times duplication [i] based on digital (177, 198, 838881)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (160, 181, 838860)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
- digital (7, 17, 21)-net over F4, using
- (u, u+v)-construction [i] based on
(178, 178+21, 4968117)-Net over F4 — Digital
Digital (178, 199, 4968117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 4968117, F4, 21) (dual of [4968117, 4967918, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, large, F4, 21) (dual of [large, large−199, 22]-code), using
- 18 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 18 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, large, F4, 21) (dual of [large, large−199, 22]-code), using
(178, 178+21, large)-Net in Base 4 — Upper bound on s
There is no (178, 199, large)-net in base 4, because
- 19 times m-reduction [i] would yield (178, 180, large)-net in base 4, but