Best Known (225, 257, s)-Nets in Base 4
(225, 257, 65545)-Net over F4 — Constructive and digital
Digital (225, 257, 65545)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (208, 240, 65536)-net over F4, using
- net defined by OOA [i] based on linear OOA(4240, 65536, F4, 32, 32) (dual of [(65536, 32), 2096912, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(4240, 1048576, F4, 32) (dual of [1048576, 1048336, 33]-code), using
- 1 times truncation [i] based on linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using
- OA 16-folding and stacking [i] based on linear OA(4240, 1048576, F4, 32) (dual of [1048576, 1048336, 33]-code), using
- net defined by OOA [i] based on linear OOA(4240, 65536, F4, 32, 32) (dual of [(65536, 32), 2096912, 33]-NRT-code), using
- digital (1, 17, 9)-net over F4, using
(225, 257, 551096)-Net over F4 — Digital
Digital (225, 257, 551096)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4257, 551096, F4, 32) (dual of [551096, 550839, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4257, 1048653, F4, 32) (dual of [1048653, 1048396, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(416, 76, F4, 6) (dual of [76, 60, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 85, F4, 6) (dual of [85, 69, 7]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4257, 1048653, F4, 32) (dual of [1048653, 1048396, 33]-code), using
(225, 257, large)-Net in Base 4 — Upper bound on s
There is no (225, 257, large)-net in base 4, because
- 30 times m-reduction [i] would yield (225, 227, large)-net in base 4, but