Best Known (19, 27, s)-Nets in Base 4
(19, 27, 240)-Net over F4 — Constructive and digital
Digital (19, 27, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 9, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
(19, 27, 280)-Net over F4 — Digital
Digital (19, 27, 280)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(427, 280, F4, 8) (dual of [280, 253, 9]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(426, 268, F4, 8) (dual of [268, 242, 9]-code), using
- construction XX applied to C1 = C([253,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,5]) [i] based on
- linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−2,−1,…,4}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(417, 255, F4, 6) (dual of [255, 238, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 255, F4, 8) (dual of [255, 230, 9]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−2,−1,…,5}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(413, 255, F4, 5) (dual of [255, 242, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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, 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([253,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([253,5]) [i] based on
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(426, 268, F4, 8) (dual of [268, 242, 9]-code), using
(19, 27, 8544)-Net in Base 4 — Upper bound on s
There is no (19, 27, 8545)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 18017 639013 753256 > 427 [i]