Best Known (138, 138+32, s)-Nets in Base 4
(138, 138+32, 1126)-Net over F4 — Constructive and digital
Digital (138, 170, 1126)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 42, 98)-net over F4, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- digital (96, 128, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 257)-net over F256, using
- digital (26, 42, 98)-net over F4, using
(138, 138+32, 9868)-Net over F4 — Digital
Digital (138, 170, 9868)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4170, 9868, F4, 32) (dual of [9868, 9698, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4170, 16399, F4, 32) (dual of [16399, 16229, 33]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4170, 16399, F4, 32) (dual of [16399, 16229, 33]-code), using
(138, 138+32, 5653324)-Net in Base 4 — Upper bound on s
There is no (138, 170, 5653325)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2 239749 787428 884100 448117 739880 177341 548904 874149 613033 691889 454554 485285 386780 779483 402069 396101 608436 > 4170 [i]