Best Known (28−7, 28, s)-Nets in Base 4
(28−7, 28, 1028)-Net over F4 — Constructive and digital
Digital (21, 28, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 7, 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
(28−7, 28, 1044)-Net over F4 — Digital
Digital (21, 28, 1044)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(428, 1044, F4, 7) (dual of [1044, 1016, 8]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] based on linear OA(427, 1036, F4, 7) (dual of [1036, 1009, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(426, 1024, F4, 7) (dual of [1024, 998, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 7 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0) [i] based on linear OA(427, 1036, F4, 7) (dual of [1036, 1009, 8]-code), using
(28−7, 28, 158780)-Net in Base 4 — Upper bound on s
There is no (21, 28, 158781)-net in base 4, because
- 1 times m-reduction [i] would yield (21, 27, 158781)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 18014 595837 483484 > 427 [i]