Best Known (145, 145+31, s)-Nets in Base 4
(145, 145+31, 1272)-Net over F4 — Constructive and digital
Digital (145, 176, 1272)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (33, 48, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
- digital (97, 128, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 32, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 32, 258)-net over F256, using
- digital (33, 48, 240)-net over F4, using
(145, 145+31, 16440)-Net over F4 — Digital
Digital (145, 176, 16440)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4176, 16440, F4, 31) (dual of [16440, 16264, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
(145, 145+31, large)-Net in Base 4 — Upper bound on s
There is no (145, 176, large)-net in base 4, because
- 29 times m-reduction [i] would yield (145, 147, large)-net in base 4, but