Best Known (179−37, 179, s)-Nets in Base 4
(179−37, 179, 1058)-Net over F4 — Constructive and digital
Digital (142, 179, 1058)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (13, 31, 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
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- digital (111, 148, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 37, 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, 37, 257)-net over F256, using
- digital (13, 31, 30)-net over F4, using
(179−37, 179, 4726)-Net over F4 — Digital
Digital (142, 179, 4726)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4179, 4726, F4, 37) (dual of [4726, 4547, 38]-code), using
- 608 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0, 1, 75 times 0, 1, 101 times 0, 1, 129 times 0, 1, 154 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(41, 7, F4, 1) (dual of [7, 6, 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(36) ⊂ Ce(34) [i] based on
- 608 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0, 1, 75 times 0, 1, 101 times 0, 1, 129 times 0, 1, 154 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
(179−37, 179, 2263134)-Net in Base 4 — Upper bound on s
There is no (142, 179, 2263135)-net in base 4, because
- 1 times m-reduction [i] would yield (142, 178, 2263135)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 146784 416602 430346 691696 022070 971014 054826 015720 548149 891986 478197 749237 426459 106949 195001 933082 469956 748973 > 4178 [i]