Best Known (257−33, 257, s)-Nets in Base 4
(257−33, 257, 65541)-Net over F4 — Constructive and digital
Digital (224, 257, 65541)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 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 (208, 241, 65536)-net over F4, using
- net defined by OOA [i] based on linear OOA(4241, 65536, F4, 33, 33) (dual of [(65536, 33), 2162447, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [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]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using
- net defined by OOA [i] based on linear OOA(4241, 65536, F4, 33, 33) (dual of [(65536, 33), 2162447, 34]-NRT-code), using
- digital (0, 16, 5)-net over F4, using
(257−33, 257, 524326)-Net over F4 — Digital
Digital (224, 257, 524326)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4257, 524326, F4, 2, 33) (dual of [(524326, 2), 1048395, 34]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4256, 524326, F4, 2, 33) (dual of [(524326, 2), 1048396, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4256, 1048652, F4, 33) (dual of [1048652, 1048396, 34]-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(415, 75, F4, 7) (dual of [75, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- OOA 2-folding [i] based on linear OA(4256, 1048652, F4, 33) (dual of [1048652, 1048396, 34]-code), using
- 41 times duplication [i] based on linear OOA(4256, 524326, F4, 2, 33) (dual of [(524326, 2), 1048396, 34]-NRT-code), using
(257−33, 257, large)-Net in Base 4 — Upper bound on s
There is no (224, 257, large)-net in base 4, because
- 31 times m-reduction [i] would yield (224, 226, large)-net in base 4, but