Best Known (134−20, 134, s)-Nets in Base 4
(134−20, 134, 6568)-Net over F4 — Constructive and digital
Digital (114, 134, 6568)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 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 (100, 120, 6553)-net over F4, using
- net defined by OOA [i] based on linear OOA(4120, 6553, F4, 20, 20) (dual of [(6553, 20), 130940, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4120, 65530, F4, 20) (dual of [65530, 65410, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4120, 65535, F4, 20) (dual of [65535, 65415, 21]-code), using
- 1 times truncation [i] based on linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times truncation [i] based on linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4120, 65535, F4, 20) (dual of [65535, 65415, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4120, 65530, F4, 20) (dual of [65530, 65410, 21]-code), using
- net defined by OOA [i] based on linear OOA(4120, 6553, F4, 20, 20) (dual of [(6553, 20), 130940, 21]-NRT-code), using
- digital (4, 14, 15)-net over F4, using
(134−20, 134, 65597)-Net over F4 — Digital
Digital (114, 134, 65597)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4134, 65597, F4, 20) (dual of [65597, 65463, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(12) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(20) ⊂ Ce(12) [i] based on
(134−20, 134, large)-Net in Base 4 — Upper bound on s
There is no (114, 134, large)-net in base 4, because
- 18 times m-reduction [i] would yield (114, 116, large)-net in base 4, but