Best Known (170−35, 170, s)-Nets in Base 4
(170−35, 170, 1058)-Net over F4 — Constructive and digital
Digital (135, 170, 1058)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 30, 30)-net over F4, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- F4 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- a shift-net [i]
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- digital (105, 140, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 35, 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, 35, 257)-net over F256, using
- digital (13, 30, 30)-net over F4, using
(170−35, 170, 4665)-Net over F4 — Digital
Digital (135, 170, 4665)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4170, 4665, F4, 35) (dual of [4665, 4495, 36]-code), using
- 550 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0, 1, 123 times 0, 1, 151 times 0) [i] based on linear OA(4157, 4102, F4, 35) (dual of [4102, 3945, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4151, 4096, F4, 34) (dual of [4096, 3945, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 550 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0, 1, 123 times 0, 1, 151 times 0) [i] based on linear OA(4157, 4102, F4, 35) (dual of [4102, 3945, 36]-code), using
(170−35, 170, 2312096)-Net in Base 4 — Upper bound on s
There is no (135, 170, 2312097)-net in base 4, because
- 1 times m-reduction [i] would yield (135, 169, 2312097)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 559937 183861 062238 030710 571298 737957 037272 706609 425564 646680 646157 203595 335123 181821 316777 461205 462360 > 4169 [i]