Best Known (81, 81+28, s)-Nets in Base 4
(81, 81+28, 531)-Net over F4 — Constructive and digital
Digital (81, 109, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (81, 111, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 177)-net over F64, using
(81, 81+28, 1051)-Net over F4 — Digital
Digital (81, 109, 1051)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4109, 1051, F4, 28) (dual of [1051, 942, 29]-code), using
- 24 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 13 times 0) [i] based on linear OA(4105, 1023, F4, 28) (dual of [1023, 918, 29]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 24 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 13 times 0) [i] based on linear OA(4105, 1023, F4, 28) (dual of [1023, 918, 29]-code), using
(81, 81+28, 98119)-Net in Base 4 — Upper bound on s
There is no (81, 109, 98120)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 421295 337956 415940 065781 139653 227333 602452 620047 963840 516811 352120 > 4109 [i]