Best Known (129−34, 129, s)-Nets in Base 4
(129−34, 129, 531)-Net over F4 — Constructive and digital
Digital (95, 129, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (95, 132, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 44, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 44, 177)-net over F64, using
(129−34, 129, 1045)-Net over F4 — Digital
Digital (95, 129, 1045)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4129, 1045, F4, 34) (dual of [1045, 916, 35]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(4126, 1029, F4, 34) (dual of [1029, 903, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(4126, 1024, F4, 34) (dual of [1024, 898, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4121, 1024, F4, 33) (dual of [1024, 903, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(4126, 1029, F4, 34) (dual of [1029, 903, 35]-code), using
(129−34, 129, 88578)-Net in Base 4 — Upper bound on s
There is no (95, 129, 88579)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 463185 451288 212678 745932 430513 400069 137728 346157 269297 434902 858921 379170 714560 > 4129 [i]