Best Known (247−31, 247, s)-Nets in Base 4
(247−31, 247, 69914)-Net over F4 — Constructive and digital
Digital (216, 247, 69914)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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 (200, 231, 69905)-net over F4, using
- net defined by OOA [i] based on linear OOA(4231, 69905, F4, 31, 31) (dual of [(69905, 31), 2166824, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using
- net defined by OOA [i] based on linear OOA(4231, 69905, F4, 31, 31) (dual of [(69905, 31), 2166824, 32]-NRT-code), using
- digital (1, 16, 9)-net over F4, using
(247−31, 247, 524326)-Net over F4 — Digital
Digital (216, 247, 524326)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4247, 524326, F4, 2, 31) (dual of [(524326, 2), 1048405, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4247, 1048652, F4, 31) (dual of [1048652, 1048405, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4246, 1048651, F4, 31) (dual of [1048651, 1048405, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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 Ce(30) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4246, 1048651, F4, 31) (dual of [1048651, 1048405, 32]-code), using
- OOA 2-folding [i] based on linear OA(4247, 1048652, F4, 31) (dual of [1048652, 1048405, 32]-code), using
(247−31, 247, large)-Net in Base 4 — Upper bound on s
There is no (216, 247, large)-net in base 4, because
- 29 times m-reduction [i] would yield (216, 218, large)-net in base 4, but