Best Known (27, 27+8, s)-Nets in Base 4
(27, 27+8, 1028)-Net over F4 — Constructive and digital
Digital (27, 35, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (27, 36, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
(27, 27+8, 1221)-Net over F4 — Digital
Digital (27, 35, 1221)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(435, 1221, F4, 8) (dual of [1221, 1186, 9]-code), using
- 179 step Varšamov–Edel lengthening with (ri) = (1, 13 times 0, 1, 46 times 0, 1, 117 times 0) [i] based on linear OA(432, 1039, F4, 8) (dual of [1039, 1007, 9]-code), using
- construction XX applied to C1 = C([1021,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,5]) [i] based on
- linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(421, 1023, F4, 6) (dual of [1023, 1002, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(431, 1023, F4, 8) (dual of [1023, 992, 9]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,5}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(416, 1023, F4, 5) (dual of [1023, 1007, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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, 5, F4, 0) (dual of [5, 5, 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([1021,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1021,5]) [i] based on
- 179 step Varšamov–Edel lengthening with (ri) = (1, 13 times 0, 1, 46 times 0, 1, 117 times 0) [i] based on linear OA(432, 1039, F4, 8) (dual of [1039, 1007, 9]-code), using
(27, 27+8, 136756)-Net in Base 4 — Upper bound on s
There is no (27, 35, 136757)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1180 614270 248005 249429 > 435 [i]