Best Known (140−15, 140, s)-Nets in Base 4
(140−15, 140, 1198376)-Net over F4 — Constructive and digital
Digital (125, 140, 1198376)-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 (118, 133, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- digital (0, 7, 5)-net over F4, using
(140−15, 140, 5172432)-Net over F4 — Digital
Digital (125, 140, 5172432)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4140, 5172432, F4, 15) (dual of [5172432, 5172292, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4140, large, F4, 15) (dual of [large, large−140, 16]-code), using
- 7 times code embedding in larger space [i] based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 7 times code embedding in larger space [i] based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4140, large, F4, 15) (dual of [large, large−140, 16]-code), using
(140−15, 140, large)-Net in Base 4 — Upper bound on s
There is no (125, 140, large)-net in base 4, because
- 13 times m-reduction [i] would yield (125, 127, large)-net in base 4, but