Best Known (68−18, 68, s)-Nets in Base 4
(68−18, 68, 312)-Net over F4 — Constructive and digital
Digital (50, 68, 312)-net over F4, using
- t-expansion [i] based on digital (49, 68, 312)-net over F4, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (49, 69, 312)-net over F4, using
(68−18, 68, 387)-Net in Base 4 — Constructive
(50, 68, 387)-net in base 4, using
- 1 times m-reduction [i] based on (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
- trace code for nets [i] based on (4, 23, 129)-net in base 64, using
(68−18, 68, 741)-Net over F4 — Digital
Digital (50, 68, 741)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(468, 741, F4, 18) (dual of [741, 673, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 1032, F4, 18) (dual of [1032, 964, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(466, 1024, F4, 18) (dual of [1024, 958, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(461, 1024, F4, 17) (dual of [1024, 963, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(456, 1024, F4, 15) (dual of [1024, 968, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(17) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(468, 1032, F4, 18) (dual of [1032, 964, 19]-code), using
(68−18, 68, 48917)-Net in Base 4 — Upper bound on s
There is no (50, 68, 48918)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 87117 566792 429695 478840 901017 771196 538918 > 468 [i]