Best Known (138, 138+15, s)-Nets in Base 4
(138, 138+15, 1198451)-Net over F4 — Constructive and digital
Digital (138, 153, 1198451)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 20, 80)-net over F4, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 20)-net over F4, using
- s-reduction based on digital (0, 1, s)-net over F4 with arbitrarily large s, using
- digital (1, 3, 20)-net over F4, using
- s-reduction based on digital (1, 3, 21)-net over F4, using
- digital (2, 5, 20)-net over F4, using
- s-reduction based on digital (2, 5, 41)-net over F4, using
- net defined by OOA [i] based on linear OOA(45, 41, F4, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(45, 41, F4, 2, 3) (dual of [(41, 2), 77, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(45, 41, F4, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 41)-net over F4, using
- digital (4, 11, 20)-net over F4, using
- digital (0, 1, 20)-net over F4, using
- generalized (u, u+v)-construction [i] based on
- 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 (13, 20, 80)-net over F4, using
(138, 138+15, 1198459)-Net in Base 4 — Constructive
(138, 153, 1198459)-net in base 4, using
- (u, u+v)-construction [i] based on
- (13, 20, 88)-net in base 4, using
- trace code for nets [i] based on (3, 10, 44)-net in base 16, using
- base change [i] based on digital (1, 8, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- base change [i] based on digital (1, 8, 44)-net over F32, using
- trace code for nets [i] based on (3, 10, 44)-net in base 16, 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
- (13, 20, 88)-net in base 4, using
(138, 138+15, large)-Net over F4 — Digital
Digital (138, 153, large)-net over F4, using
- 48 times duplication [i] based on digital (130, 145, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
(138, 138+15, large)-Net in Base 4 — Upper bound on s
There is no (138, 153, large)-net in base 4, because
- 13 times m-reduction [i] would yield (138, 140, large)-net in base 4, but