Best Known (134, 134+34, s)-Nets in Base 4
(134, 134+34, 1061)-Net over F4 — Constructive and digital
Digital (134, 168, 1061)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (15, 32, 33)-net over F4, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 15 and N(F) ≥ 33, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- digital (102, 136, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 34, 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, 34, 257)-net over F256, using
- digital (15, 32, 33)-net over F4, using
(134, 134+34, 5117)-Net over F4 — Digital
Digital (134, 168, 5117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4168, 5117, F4, 34) (dual of [5117, 4949, 35]-code), using
- 998 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 20 times 0, 1, 33 times 0, 1, 50 times 0, 1, 73 times 0, 1, 101 times 0, 1, 133 times 0, 1, 161 times 0, 1, 184 times 0, 1, 201 times 0) [i] based on linear OA(4151, 4102, F4, 34) (dual of [4102, 3951, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 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(4145, 4096, F4, 33) (dual of [4096, 3951, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 998 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 20 times 0, 1, 33 times 0, 1, 50 times 0, 1, 73 times 0, 1, 101 times 0, 1, 133 times 0, 1, 161 times 0, 1, 184 times 0, 1, 201 times 0) [i] based on linear OA(4151, 4102, F4, 34) (dual of [4102, 3951, 35]-code), using
(134, 134+34, 2131034)-Net in Base 4 — Upper bound on s
There is no (134, 168, 2131035)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 139984 537714 493677 919803 453697 898358 758538 657031 974344 793119 838198 940294 935249 544046 216536 851576 293238 > 4168 [i]