Best Known (117−31, 117, s)-Nets in Base 4
(117−31, 117, 531)-Net over F4 — Constructive and digital
Digital (86, 117, 531)-net over F4, using
- t-expansion [i] based on digital (85, 117, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 39, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 39, 177)-net over F64, using
(117−31, 117, 974)-Net over F4 — Digital
Digital (86, 117, 974)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4117, 974, F4, 31) (dual of [974, 857, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4117, 1035, F4, 31) (dual of [1035, 918, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4116, 1024, F4, 31) (dual of [1024, 908, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4117, 1035, F4, 31) (dual of [1035, 918, 32]-code), using
(117−31, 117, 96944)-Net in Base 4 — Upper bound on s
There is no (86, 117, 96945)-net in base 4, because
- 1 times m-reduction [i] would yield (86, 116, 96945)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6902 504685 941271 708386 552205 213985 890512 441326 860849 878089 633365 518512 > 4116 [i]