Best Known (48−13, 48, s)-Nets in Base 4
(48−13, 48, 312)-Net over F4 — Constructive and digital
Digital (35, 48, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 16, 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
(48−13, 48, 387)-Net in Base 4 — Constructive
(35, 48, 387)-net in base 4, using
- trace code for nets [i] based on (3, 16, 129)-net in base 64, using
- 5 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
- 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
(48−13, 48, 604)-Net over F4 — Digital
Digital (35, 48, 604)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(448, 604, F4, 13) (dual of [604, 556, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(448, 1032, F4, 13) (dual of [1032, 984, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(441, 1024, F4, 11) (dual of [1024, 983, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(436, 1024, F4, 10) (dual of [1024, 988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(448, 1032, F4, 13) (dual of [1032, 984, 14]-code), using
(48−13, 48, 51903)-Net in Base 4 — Upper bound on s
There is no (35, 48, 51904)-net in base 4, because
- 1 times m-reduction [i] would yield (35, 47, 51904)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 19807 328282 281668 573294 920977 > 447 [i]