Best Known (213−39, 213, s)-Nets in Base 4
(213−39, 213, 1539)-Net over F4 — Constructive and digital
Digital (174, 213, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (174, 219, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 73, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 73, 513)-net over F64, using
(213−39, 213, 13724)-Net over F4 — Digital
Digital (174, 213, 13724)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4213, 13724, F4, 39) (dual of [13724, 13511, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4213, 16421, F4, 39) (dual of [16421, 16208, 40]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4211, 16419, F4, 39) (dual of [16419, 16208, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- linear OA(4204, 16384, F4, 39) (dual of [16384, 16180, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4211, 16419, F4, 39) (dual of [16419, 16208, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4213, 16421, F4, 39) (dual of [16421, 16208, 40]-code), using
(213−39, 213, large)-Net in Base 4 — Upper bound on s
There is no (174, 213, large)-net in base 4, because
- 37 times m-reduction [i] would yield (174, 176, large)-net in base 4, but