Best Known (171, 171+38, s)-Nets in Base 4
(171, 171+38, 1539)-Net over F4 — Constructive and digital
Digital (171, 209, 1539)-net over F4, using
- t-expansion [i] based on digital (170, 209, 1539)-net over F4, using
- 4 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 71, 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, 71, 513)-net over F64, using
- 4 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
(171, 171+38, 14299)-Net over F4 — Digital
Digital (171, 209, 14299)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 14299, F4, 38) (dual of [14299, 14090, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 16424, F4, 38) (dual of [16424, 16215, 39]-code), using
- 5 times code embedding in larger space [i] based on linear OA(4204, 16419, F4, 38) (dual of [16419, 16215, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(32) [i] based on
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(37) ⊂ Ce(32) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(4204, 16419, F4, 38) (dual of [16419, 16215, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 16424, F4, 38) (dual of [16424, 16215, 39]-code), using
(171, 171+38, large)-Net in Base 4 — Upper bound on s
There is no (171, 209, large)-net in base 4, because
- 36 times m-reduction [i] would yield (171, 173, large)-net in base 4, but