Best Known (119, 144, s)-Nets in Base 4
(119, 144, 1382)-Net over F4 — Constructive and digital
Digital (119, 144, 1382)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (102, 127, 1365)-net over F4, using
- net defined by OOA [i] based on linear OOA(4127, 1365, F4, 25, 25) (dual of [(1365, 25), 33998, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4127, 16381, F4, 25) (dual of [16381, 16254, 26]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4127, 16381, F4, 25) (dual of [16381, 16254, 26]-code), using
- net defined by OOA [i] based on linear OOA(4127, 1365, F4, 25, 25) (dual of [(1365, 25), 33998, 26]-NRT-code), using
- digital (5, 17, 17)-net over F4, using
(119, 144, 1415)-Net in Base 4 — Constructive
(119, 144, 1415)-net in base 4, using
- 42 times duplication [i] based on (117, 142, 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 (75, 100, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 25, 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, 25, 257)-net over F256, using
- (30, 42, 387)-net in base 4, using
- (u, u+v)-construction [i] based on
(119, 144, 16446)-Net over F4 — Digital
Digital (119, 144, 16446)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4144, 16446, F4, 25) (dual of [16446, 16302, 26]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4141, 16441, F4, 25) (dual of [16441, 16300, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [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]
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4141, 16443, F4, 23) (dual of [16443, 16302, 24]-code), using Gilbert–Varšamov bound and bm = 4141 > Vbs−1(k−1) = 15523 617466 758170 786692 576587 377024 767271 878985 550153 593645 355557 347704 545671 343524 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4141, 16441, F4, 25) (dual of [16441, 16300, 26]-code), using
- construction X with Varšamov bound [i] based on
(119, 144, large)-Net in Base 4 — Upper bound on s
There is no (119, 144, large)-net in base 4, because
- 23 times m-reduction [i] would yield (119, 121, large)-net in base 4, but