Best Known (97−12, 97, s)-Nets in Base 4
(97−12, 97, 174771)-Net over F4 — Constructive and digital
Digital (85, 97, 174771)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (78, 90, 174762)-net over F4, using
- net defined by OOA [i] based on linear OOA(490, 174762, F4, 12, 12) (dual of [(174762, 12), 2097054, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(490, 1048572, F4, 12) (dual of [1048572, 1048482, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 1048575, F4, 12) (dual of [1048575, 1048485, 13]-code), using
- 1 times truncation [i] based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 1 times truncation [i] based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 1048575, F4, 12) (dual of [1048575, 1048485, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(490, 1048572, F4, 12) (dual of [1048572, 1048482, 13]-code), using
- net defined by OOA [i] based on linear OOA(490, 174762, F4, 12, 12) (dual of [(174762, 12), 2097054, 13]-NRT-code), using
- digital (1, 7, 9)-net over F4, using
(97−12, 97, 909133)-Net over F4 — Digital
Digital (85, 97, 909133)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(497, 909133, F4, 12) (dual of [909133, 909036, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(497, 1048612, F4, 12) (dual of [1048612, 1048515, 13]-code), using
- 2 times code embedding in larger space [i] based on linear OA(495, 1048610, F4, 12) (dual of [1048610, 1048515, 13]-code), using
- 1 times truncation [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(495, 1048610, F4, 12) (dual of [1048610, 1048515, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(497, 1048612, F4, 12) (dual of [1048612, 1048515, 13]-code), using
(97−12, 97, large)-Net in Base 4 — Upper bound on s
There is no (85, 97, large)-net in base 4, because
- 10 times m-reduction [i] would yield (85, 87, large)-net in base 4, but