Best Known (67−18, 67, s)-Nets in Base 4
(67−18, 67, 312)-Net over F4 — Constructive and digital
Digital (49, 67, 312)-net over F4, using
- 2 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
(67−18, 67, 387)-Net in Base 4 — Constructive
(49, 67, 387)-net in base 4, using
- 44 times duplication [i] based on (45, 63, 387)-net in base 4, using
- trace code for nets [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 129)-net over F128, using
- trace code for nets [i] based on (3, 21, 129)-net in base 64, using
(67−18, 67, 678)-Net over F4 — Digital
Digital (49, 67, 678)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(467, 678, F4, 18) (dual of [678, 611, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(467, 1034, F4, 18) (dual of [1034, 967, 19]-code), using
- construction XX applied to C1 = C([325,341]), C2 = C([327,342]), C3 = C1 + C2 = C([327,341]), and C∩ = C1 ∩ C2 = C([325,342]) [i] based on
- linear OA(461, 1023, F4, 17) (dual of [1023, 962, 18]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {325,326,…,341}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(461, 1023, F4, 16) (dual of [1023, 962, 17]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {327,328,…,342}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(466, 1023, F4, 18) (dual of [1023, 957, 19]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {325,326,…,342}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(456, 1023, F4, 15) (dual of [1023, 967, 16]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {327,328,…,341}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 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
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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
- construction XX applied to C1 = C([325,341]), C2 = C([327,342]), C3 = C1 + C2 = C([327,341]), and C∩ = C1 ∩ C2 = C([325,342]) [i] based on
- discarding factors / shortening the dual code based on linear OA(467, 1034, F4, 18) (dual of [1034, 967, 19]-code), using
(67−18, 67, 41933)-Net in Base 4 — Upper bound on s
There is no (49, 67, 41934)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 21780 971509 178035 562025 952761 241357 325413 > 467 [i]