Best Known (178, 178+38, s)-Nets in Base 4
(178, 178+38, 1539)-Net over F4 — Constructive and digital
Digital (178, 216, 1539)-net over F4, using
- 9 times m-reduction [i] based on digital (178, 225, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
(178, 178+38, 16453)-Net over F4 — Digital
Digital (178, 216, 16453)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4216, 16453, F4, 38) (dual of [16453, 16237, 39]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4215, 16451, F4, 38) (dual of [16451, 16236, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [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(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(418, 67, F4, 8) (dual of [67, 49, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(418, 68, F4, 8) (dual of [68, 50, 9]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- linear OA(4215, 16452, F4, 37) (dual of [16452, 16237, 38]-code), using Gilbert–Varšamov bound and bm = 4215 > Vbs−1(k−1) = 23 573657 126516 550663 491782 658670 762352 248556 076389 724402 882582 868695 140782 739104 324904 495368 479772 912485 429111 101385 092733 537200 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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(4215, 16451, F4, 38) (dual of [16451, 16236, 39]-code), using
- construction X with Varšamov bound [i] based on
(178, 178+38, large)-Net in Base 4 — Upper bound on s
There is no (178, 216, large)-net in base 4, because
- 36 times m-reduction [i] would yield (178, 180, large)-net in base 4, but