Best Known (237−42, 237, s)-Nets in Base 4
(237−42, 237, 1556)-Net over F4 — Constructive and digital
Digital (195, 237, 1556)-net over F4, using
- 41 times duplication [i] based on digital (194, 236, 1556)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 26, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (168, 210, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 70, 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, 70, 513)-net over F64, using
- digital (5, 26, 17)-net over F4, using
- (u, u+v)-construction [i] based on
(237−42, 237, 16453)-Net over F4 — Digital
Digital (195, 237, 16453)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4237, 16453, F4, 42) (dual of [16453, 16216, 43]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4236, 16451, F4, 42) (dual of [16451, 16215, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(32) [i] based on
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [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(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(41) ⊂ Ce(32) [i] based on
- linear OA(4236, 16452, F4, 41) (dual of [16452, 16216, 42]-code), using Gilbert–Varšamov bound and bm = 4236 > Vbs−1(k−1) = 63 189110 647489 946867 820065 711473 656328 098403 678493 572772 614151 553585 495328 775078 822302 716940 181130 885155 012569 782978 494346 738607 358686 115560 [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(4236, 16451, F4, 42) (dual of [16451, 16215, 43]-code), using
- construction X with Varšamov bound [i] based on
(237−42, 237, large)-Net in Base 4 — Upper bound on s
There is no (195, 237, large)-net in base 4, because
- 40 times m-reduction [i] would yield (195, 197, large)-net in base 4, but