Best Known (217, 217+35, s)-Nets in Base 4
(217, 217+35, 15425)-Net over F4 — Constructive and digital
Digital (217, 252, 15425)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 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 (200, 235, 15420)-net over F4, using
- net defined by OOA [i] based on linear OOA(4235, 15420, F4, 35, 35) (dual of [(15420, 35), 539465, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4235, 262141, F4, 35) (dual of [262141, 261906, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- discarding factors / shortening the dual code based on linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4235, 262141, F4, 35) (dual of [262141, 261906, 36]-code), using
- net defined by OOA [i] based on linear OOA(4235, 15420, F4, 35, 35) (dual of [(15420, 35), 539465, 36]-NRT-code), using
- digital (0, 17, 5)-net over F4, using
(217, 217+35, 166521)-Net over F4 — Digital
Digital (217, 252, 166521)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4252, 166521, F4, 35) (dual of [166521, 166269, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4252, 262213, F4, 35) (dual of [262213, 261961, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(4235, 262145, F4, 35) (dual of [262145, 261910, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4181, 262145, F4, 27) (dual of [262145, 261964, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(417, 68, F4, 7) (dual of [68, 51, 8]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- linear OA(49, 17, F4, 7) (dual of [17, 8, 8]-code), using
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4252, 262213, F4, 35) (dual of [262213, 261961, 36]-code), using
(217, 217+35, large)-Net in Base 4 — Upper bound on s
There is no (217, 252, large)-net in base 4, because
- 33 times m-reduction [i] would yield (217, 219, large)-net in base 4, but