Best Known (115, 115+24, s)-Nets in Base 4
(115, 115+24, 1374)-Net over F4 — Constructive and digital
Digital (115, 139, 1374)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (102, 126, 1365)-net over F4, using
- net defined by OOA [i] based on linear OOA(4126, 1365, F4, 24, 24) (dual of [(1365, 24), 32634, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4126, 16380, F4, 24) (dual of [16380, 16254, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 16384, F4, 24) (dual of [16384, 16258, 25]-code), using
- 1 times truncation [i] based on linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 16384, F4, 24) (dual of [16384, 16258, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4126, 16380, F4, 24) (dual of [16380, 16254, 25]-code), using
- net defined by OOA [i] based on linear OOA(4126, 1365, F4, 24, 24) (dual of [(1365, 24), 32634, 25]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(115, 115+24, 1415)-Net in Base 4 — Constructive
(115, 139, 1415)-net in base 4, using
- 41 times duplication [i] based on (114, 138, 1415)-net in base 4, using
- (u, u+v)-construction [i] based on
- (30, 42, 387)-net in base 4, using
- trace code for nets [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 129)-net over F128, using
- trace code for nets [i] based on (2, 14, 129)-net in base 64, using
- digital (72, 96, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- (30, 42, 387)-net in base 4, using
- (u, u+v)-construction [i] based on
(115, 115+24, 16433)-Net over F4 — Digital
Digital (115, 139, 16433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4139, 16433, F4, 24) (dual of [16433, 16294, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(410, 45, F4, 5) (dual of [45, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4137, 16431, F4, 23) (dual of [16431, 16294, 24]-code), using Gilbert–Varšamov bound and bm = 4137 > Vbs−1(k−1) = 15276 112369 899042 865413 274680 127703 757431 152895 921510 109551 812602 894225 601989 727384 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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
- linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
- construction X with Varšamov bound [i] based on
(115, 115+24, large)-Net in Base 4 — Upper bound on s
There is no (115, 139, large)-net in base 4, because
- 22 times m-reduction [i] would yield (115, 117, large)-net in base 4, but