Best Known (206, 206+49, s)-Nets in Base 4
(206, 206+49, 1544)-Net over F4 — Constructive and digital
Digital (206, 255, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 24, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (182, 231, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 77, 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, 77, 513)-net over F64, using
- digital (0, 24, 5)-net over F4, using
(206, 206+49, 10944)-Net over F4 — Digital
Digital (206, 255, 10944)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4255, 10944, F4, 49) (dual of [10944, 10689, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4255, 16394, F4, 49) (dual of [16394, 16139, 50]-code), using
- construction XX applied to Ce(48) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- linear OA(4253, 16384, F4, 49) (dual of [16384, 16131, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(4246, 16384, F4, 47) (dual of [16384, 16138, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(4239, 16384, F4, 46) (dual of [16384, 16145, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(48) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(4255, 16394, F4, 49) (dual of [16394, 16139, 50]-code), using
(206, 206+49, 7692047)-Net in Base 4 — Upper bound on s
There is no (206, 255, 7692048)-net in base 4, because
- 1 times m-reduction [i] would yield (206, 254, 7692048)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 837 989394 635376 900500 977607 997497 705004 934825 528314 301136 850895 119846 710236 947021 310231 075222 858160 282101 651807 924294 844882 023087 460686 265015 072988 791056 > 4254 [i]