Best Known (138, 138+33, s)-Nets in Base 4
(138, 138+33, 1104)-Net over F4 — Constructive and digital
Digital (138, 171, 1104)-net over F4, using
- 41 times duplication [i] based on digital (137, 170, 1104)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (22, 38, 76)-net over F4, using
- trace code for nets [i] based on digital (3, 19, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 19, 38)-net over F16, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- digital (22, 38, 76)-net over F4, using
- (u, u+v)-construction [i] based on
(138, 138+33, 8265)-Net over F4 — Digital
Digital (138, 171, 8265)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4171, 8265, F4, 33) (dual of [8265, 8094, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 16394, F4, 33) (dual of [16394, 16223, 34]-code), using
- construction XX applied to Ce(32) ⊂ Ce(30) ⊂ 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(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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, 9, F4, 1) (dual of [9, 8, 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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(32) ⊂ Ce(30) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 16394, F4, 33) (dual of [16394, 16223, 34]-code), using
(138, 138+33, 5653324)-Net in Base 4 — Upper bound on s
There is no (138, 171, 5653325)-net in base 4, because
- 1 times m-reduction [i] would yield (138, 170, 5653325)-net in base 4, but
- 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]