Best Known (110, 110+13, s)-Nets in Base 4
(110, 110+13, 1398148)-Net over F4 — Constructive and digital
Digital (110, 123, 1398148)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (8, 14, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 7, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 7, 24)-net over F16, using
- digital (96, 109, 1398100)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
- digital (8, 14, 48)-net over F4, using
(110, 110+13, 7785474)-Net over F4 — Digital
Digital (110, 123, 7785474)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4123, 7785474, F4, 13) (dual of [7785474, 7785351, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4123, large, F4, 13) (dual of [large, large−123, 14]-code), using
- 14 times code embedding in larger space [i] based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 14 times code embedding in larger space [i] based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4123, large, F4, 13) (dual of [large, large−123, 14]-code), using
(110, 110+13, large)-Net in Base 4 — Upper bound on s
There is no (110, 123, large)-net in base 4, because
- 11 times m-reduction [i] would yield (110, 112, large)-net in base 4, but