Best Known (52−14, 52, s)-Nets in Base 4
(52−14, 52, 312)-Net over F4 — Constructive and digital
Digital (38, 52, 312)-net over F4, using
- 41 times duplication [i] based on digital (37, 51, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 17, 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, 17, 104)-net over F64, using
(52−14, 52, 387)-Net in Base 4 — Constructive
(38, 52, 387)-net in base 4, using
- 41 times duplication [i] based on (37, 51, 387)-net in base 4, using
- trace code for nets [i] based on (3, 17, 129)-net in base 64, using
- 4 times m-reduction [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
- 4 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- trace code for nets [i] based on (3, 17, 129)-net in base 64, using
(52−14, 52, 630)-Net over F4 — Digital
Digital (38, 52, 630)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(452, 630, F4, 14) (dual of [630, 578, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(452, 1034, F4, 14) (dual of [1034, 982, 15]-code), using
- construction XX applied to C1 = C([329,341]), C2 = C([331,342]), C3 = C1 + C2 = C([331,341]), and C∩ = C1 ∩ C2 = C([329,342]) [i] based on
- linear OA(446, 1023, F4, 13) (dual of [1023, 977, 14]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {329,330,…,341}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(446, 1023, F4, 12) (dual of [1023, 977, 13]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {331,332,…,342}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(451, 1023, F4, 14) (dual of [1023, 972, 15]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {329,330,…,342}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(441, 1023, F4, 11) (dual of [1023, 982, 12]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {331,332,…,341}, and designed minimum distance d ≥ |I|+1 = 12 [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([329,341]), C2 = C([331,342]), C3 = C1 + C2 = C([331,341]), and C∩ = C1 ∩ C2 = C([329,342]) [i] based on
- discarding factors / shortening the dual code based on linear OA(452, 1034, F4, 14) (dual of [1034, 982, 15]-code), using
(52−14, 52, 33432)-Net in Base 4 — Upper bound on s
There is no (38, 52, 33433)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 20 282924 551218 469073 070087 311896 > 452 [i]