Best Known (69−19, 69, s)-Nets in Base 4
(69−19, 69, 312)-Net over F4 — Constructive and digital
Digital (50, 69, 312)-net over F4, using
- t-expansion [i] based on digital (49, 69, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 23, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 23, 104)-net over F64, using
(69−19, 69, 387)-Net in Base 4 — Constructive
(50, 69, 387)-net in base 4, using
- trace code for nets [i] based on (4, 23, 129)-net in base 64, using
- 5 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- 5 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
(69−19, 69, 534)-Net over F4 — Digital
Digital (50, 69, 534)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(469, 534, F4, 19) (dual of [534, 465, 20]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(468, 516, F4, 19) (dual of [516, 448, 20]-code), using
- trace code [i] based on linear OA(1634, 258, F16, 19) (dual of [258, 224, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(1634, 256, F16, 19) (dual of [256, 222, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1632, 256, F16, 18) (dual of [256, 224, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(1634, 258, F16, 19) (dual of [258, 224, 20]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(468, 516, F4, 19) (dual of [516, 448, 20]-code), using
(69−19, 69, 48917)-Net in Base 4 — Upper bound on s
There is no (50, 69, 48918)-net in base 4, because
- 1 times m-reduction [i] would yield (50, 68, 48918)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 87117 566792 429695 478840 901017 771196 538918 > 468 [i]