Best Known (178−25, 178, s)-Nets in Base 4
(178−25, 178, 21859)-Net over F4 — Constructive and digital
Digital (153, 178, 21859)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (138, 163, 21845)-net over F4, using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- digital (3, 15, 14)-net over F4, using
(178−25, 178, 135058)-Net over F4 — Digital
Digital (153, 178, 135058)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 135058, F4, 25) (dual of [135058, 134880, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, 262163, F4, 25) (dual of [262163, 261985, 26]-code), using
- (u, u+v)-construction [i] based on
- linear OA(415, 18, F4, 12) (dual of [18, 3, 13]-code), using
- 3 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- Simplex code S(3,4) [i]
- 3 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(415, 18, F4, 12) (dual of [18, 3, 13]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4178, 262163, F4, 25) (dual of [262163, 261985, 26]-code), using
(178−25, 178, large)-Net in Base 4 — Upper bound on s
There is no (153, 178, large)-net in base 4, because
- 23 times m-reduction [i] would yield (153, 155, large)-net in base 4, but