Best Known (157−12, 157, s)-Nets in Base 4
(157−12, 157, 5592743)-Net over F4 — Constructive and digital
Digital (145, 157, 5592743)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (15, 21, 343)-net over F4, using
- net defined by OOA [i] based on linear OOA(421, 343, F4, 6, 6) (dual of [(343, 6), 2037, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(421, 1029, F4, 6) (dual of [1029, 1008, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(421, 1024, F4, 6) (dual of [1024, 1003, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(421, 1029, F4, 6) (dual of [1029, 1008, 7]-code), using
- net defined by OOA [i] based on linear OOA(421, 343, F4, 6, 6) (dual of [(343, 6), 2037, 7]-NRT-code), using
- digital (124, 136, 5592400)-net over F4, using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- digital (15, 21, 343)-net over F4, using
(157−12, 157, 5592787)-Net in Base 4 — Constructive
(145, 157, 5592787)-net in base 4, using
- (u, u+v)-construction [i] based on
- (15, 21, 387)-net in base 4, using
- trace code for nets [i] based on (1, 7, 129)-net in base 64, using
- base change [i] based on digital (0, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 6, 129)-net over F128, using
- trace code for nets [i] based on (1, 7, 129)-net in base 64, using
- digital (124, 136, 5592400)-net over F4, using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- trace code for nets [i] based on digital (56, 68, 2796200)-net over F16, using
- (15, 21, 387)-net in base 4, using
(157−12, 157, large)-Net over F4 — Digital
Digital (145, 157, large)-net over F4, using
- 42 times duplication [i] based on digital (143, 155, large)-net over F4, using
- t-expansion [i] based on digital (139, 155, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- t-expansion [i] based on digital (139, 155, large)-net over F4, using
(157−12, 157, large)-Net in Base 4 — Upper bound on s
There is no (145, 157, large)-net in base 4, because
- 10 times m-reduction [i] would yield (145, 147, large)-net in base 4, but