Best Known (251−29, 251, s)-Nets in Base 4
(251−29, 251, 299610)-Net over F4 — Constructive and digital
Digital (222, 251, 299610)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (203, 232, 299593)-net over F4, using
- net defined by OOA [i] based on linear OOA(4232, 299593, F4, 29, 29) (dual of [(299593, 29), 8687965, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4232, 4194303, F4, 29) (dual of [4194303, 4194071, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4232, 4194303, F4, 29) (dual of [4194303, 4194071, 30]-code), using
- net defined by OOA [i] based on linear OOA(4232, 299593, F4, 29, 29) (dual of [(299593, 29), 8687965, 30]-NRT-code), using
- digital (5, 19, 17)-net over F4, using
(251−29, 251, 2051397)-Net over F4 — Digital
Digital (222, 251, 2051397)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4251, 2051397, F4, 2, 29) (dual of [(2051397, 2), 4102543, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4251, 2097194, F4, 2, 29) (dual of [(2097194, 2), 4194137, 30]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4250, 2097194, F4, 2, 29) (dual of [(2097194, 2), 4194138, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4250, 4194388, F4, 29) (dual of [4194388, 4194138, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(418, 84, F4, 7) (dual of [84, 66, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(418, 85, F4, 7) (dual of [85, 67, 8]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(4250, 4194388, F4, 29) (dual of [4194388, 4194138, 30]-code), using
- 41 times duplication [i] based on linear OOA(4250, 2097194, F4, 2, 29) (dual of [(2097194, 2), 4194138, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4251, 2097194, F4, 2, 29) (dual of [(2097194, 2), 4194137, 30]-NRT-code), using
(251−29, 251, large)-Net in Base 4 — Upper bound on s
There is no (222, 251, large)-net in base 4, because
- 27 times m-reduction [i] would yield (222, 224, large)-net in base 4, but