Best Known (218−27, 218, s)-Nets in Base 4
(218−27, 218, 80675)-Net over F4 — Constructive and digital
Digital (191, 218, 80675)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 17, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (174, 201, 80660)-net over F4, using
- net defined by OOA [i] based on linear OOA(4201, 80660, F4, 27, 27) (dual of [(80660, 27), 2177619, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4201, 1048581, F4, 27) (dual of [1048581, 1048380, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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 X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4201, 1048581, F4, 27) (dual of [1048581, 1048380, 28]-code), using
- net defined by OOA [i] based on linear OOA(4201, 80660, F4, 27, 27) (dual of [(80660, 27), 2177619, 28]-NRT-code), using
- digital (4, 17, 15)-net over F4, using
(218−27, 218, 570651)-Net over F4 — Digital
Digital (191, 218, 570651)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4218, 570651, F4, 27) (dual of [570651, 570433, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 1048598, F4, 27) (dual of [1048598, 1048380, 28]-code), using
- (u, u+v)-construction [i] based on
- linear OA(417, 21, F4, 13) (dual of [21, 4, 14]-code), using
- 2 times truncation [i] based on linear OA(419, 23, F4, 15) (dual of [23, 4, 16]-code), using
- construction X applied to C1 ⊂ C2 with C1 a [17,1,16]-code [i] based on
- 2 times truncation [i] based on linear OA(419, 23, F4, 15) (dual of [23, 4, 16]-code), using
- linear OA(4201, 1048577, F4, 27) (dual of [1048577, 1048376, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(417, 21, F4, 13) (dual of [21, 4, 14]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 1048598, F4, 27) (dual of [1048598, 1048380, 28]-code), using
(218−27, 218, large)-Net in Base 4 — Upper bound on s
There is no (191, 218, large)-net in base 4, because
- 25 times m-reduction [i] would yield (191, 193, large)-net in base 4, but