Best Known (108−14, 108, s)-Nets in Base 4
(108−14, 108, 149803)-Net over F4 — Constructive and digital
Digital (94, 108, 149803)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (87, 101, 149798)-net over F4, using
- net defined by OOA [i] based on linear OOA(4101, 149798, F4, 14, 14) (dual of [(149798, 14), 2097071, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) [i] based on
- OA 7-folding and stacking [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- net defined by OOA [i] based on linear OOA(4101, 149798, F4, 14, 14) (dual of [(149798, 14), 2097071, 15]-NRT-code), using
- digital (0, 7, 5)-net over F4, using
(108−14, 108, 524309)-Net over F4 — Digital
Digital (94, 108, 524309)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4108, 524309, F4, 2, 14) (dual of [(524309, 2), 1048510, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4108, 1048618, F4, 14) (dual of [1048618, 1048510, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, 1048619, F4, 14) (dual of [1048619, 1048511, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(4108, 1048619, F4, 14) (dual of [1048619, 1048511, 15]-code), using
- OOA 2-folding [i] based on linear OA(4108, 1048618, F4, 14) (dual of [1048618, 1048510, 15]-code), using
(108−14, 108, large)-Net in Base 4 — Upper bound on s
There is no (94, 108, large)-net in base 4, because
- 12 times m-reduction [i] would yield (94, 96, large)-net in base 4, but