Best Known (67, 77, s)-Nets in Base 4
(67, 77, 209726)-Net over F4 — Constructive and digital
Digital (67, 77, 209726)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (61, 71, 209717)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 209717, F4, 10, 10) (dual of [(209717, 10), 2097099, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
- net defined by OOA [i] based on linear OOA(471, 209717, F4, 10, 10) (dual of [(209717, 10), 2097099, 11]-NRT-code), using
- digital (1, 6, 9)-net over F4, using
(67, 77, 657862)-Net over F4 — Digital
Digital (67, 77, 657862)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(477, 657862, F4, 10) (dual of [657862, 657785, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 1048588, F4, 10) (dual of [1048588, 1048511, 11]-code), using
- (u, u+v)-construction [i] based on
- linear OA(46, 12, F4, 5) (dual of [12, 6, 6]-code), using
- extended quadratic residue code Qe(12,4) [i]
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(46, 12, F4, 5) (dual of [12, 6, 6]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(477, 1048588, F4, 10) (dual of [1048588, 1048511, 11]-code), using
(67, 77, large)-Net in Base 4 — Upper bound on s
There is no (67, 77, large)-net in base 4, because
- 8 times m-reduction [i] would yield (67, 69, large)-net in base 4, but