Best Known (19−6, 19, s)-Nets in Base 4
(19−6, 19, 195)-Net over F4 — Constructive and digital
Digital (13, 19, 195)-net over F4, using
- 41 times duplication [i] based on digital (12, 18, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
(19−6, 19, 269)-Net over F4 — Digital
Digital (13, 19, 269)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(419, 269, F4, 6) (dual of [269, 250, 7]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(418, 264, F4, 6) (dual of [264, 246, 7]-code), using
- construction XX applied to C1 = C([81,85]), C2 = C([83,86]), C3 = C1 + C2 = C([83,85]), and C∩ = C1 ∩ C2 = C([81,86]) [i] based on
- linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {81,82,83,84,85}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(413, 255, F4, 4) (dual of [255, 242, 5]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {83,84,85,86}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(417, 255, F4, 6) (dual of [255, 238, 7]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {81,82,…,86}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(49, 255, F4, 3) (dual of [255, 246, 4]-code or 255-cap in PG(8,4)), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {83,84,85}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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, 4, F4, 0) (dual of [4, 4, 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
- construction XX applied to C1 = C([81,85]), C2 = C([83,86]), C3 = C1 + C2 = C([83,85]), and C∩ = C1 ∩ C2 = C([81,86]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(418, 264, F4, 6) (dual of [264, 246, 7]-code), using
(19−6, 19, 3936)-Net in Base 4 — Upper bound on s
There is no (13, 19, 3937)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 275023 823968 > 419 [i]