Best Known (39, 53, s)-Nets in Base 4
(39, 53, 312)-Net over F4 — Constructive and digital
Digital (39, 53, 312)-net over F4, using
- 1 times m-reduction [i] based on digital (39, 54, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 18, 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, 18, 104)-net over F64, using
(39, 53, 387)-Net in Base 4 — Constructive
(39, 53, 387)-net in base 4, using
- 1 times m-reduction [i] based on (39, 54, 387)-net in base 4, using
- trace code for nets [i] based on (3, 18, 129)-net in base 64, using
- 3 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
- 3 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- trace code for nets [i] based on (3, 18, 129)-net in base 64, using
(39, 53, 708)-Net over F4 — Digital
Digital (39, 53, 708)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 708, F4, 14) (dual of [708, 655, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(453, 1032, F4, 14) (dual of [1032, 979, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(451, 1024, F4, 14) (dual of [1024, 973, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(453, 1032, F4, 14) (dual of [1032, 979, 15]-code), using
(39, 53, 40756)-Net in Base 4 — Upper bound on s
There is no (39, 53, 40757)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 81 139341 455455 739870 255395 862320 > 453 [i]