Best Known (197, 197+45, s)-Nets in Base 4
(197, 197+45, 1548)-Net over F4 — Constructive and digital
Digital (197, 242, 1548)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 23, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- 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
- digital (1, 23, 9)-net over F4, using
(197, 197+45, 13291)-Net over F4 — Digital
Digital (197, 242, 13291)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4242, 13291, F4, 45) (dual of [13291, 13049, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4242, 16422, F4, 45) (dual of [16422, 16180, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(38) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 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(410, 38, F4, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(44) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(4242, 16422, F4, 45) (dual of [16422, 16180, 46]-code), using
(197, 197+45, large)-Net in Base 4 — Upper bound on s
There is no (197, 242, large)-net in base 4, because
- 43 times m-reduction [i] would yield (197, 199, large)-net in base 4, but