Best Known (137, 137+33, s)-Nets in Base 4
(137, 137+33, 1104)-Net over F4 — Constructive and digital
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
(137, 137+33, 8196)-Net over F4 — Digital
Digital (137, 170, 8196)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4170, 8196, F4, 2, 33) (dual of [(8196, 2), 16222, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4170, 16392, F4, 33) (dual of [16392, 16222, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [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(41, 8, F4, 1) (dual of [8, 7, 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(30) [i] based on
- OOA 2-folding [i] based on linear OA(4170, 16392, F4, 33) (dual of [16392, 16222, 34]-code), using
(137, 137+33, 5184120)-Net in Base 4 — Upper bound on s
There is no (137, 170, 5184121)-net in base 4, because
- 1 times m-reduction [i] would yield (137, 169, 5184121)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 559937 762354 510488 893978 521850 404288 808528 164912 297684 867445 457227 080243 635969 545846 797834 564811 244429 > 4169 [i]