Best Known (117−29, 117, s)-Nets in Base 4
(117−29, 117, 1028)-Net over F4 — Constructive and digital
Digital (88, 117, 1028)-net over F4, using
- 41 times duplication [i] based on digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 257)-net over F256, using
(117−29, 117, 1251)-Net over F4 — Digital
Digital (88, 117, 1251)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4117, 1251, F4, 29) (dual of [1251, 1134, 30]-code), using
- 216 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 29 times 0, 1, 39 times 0, 1, 48 times 0, 1, 55 times 0) [i] based on linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using
- an extension Ce(28) of 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]
- 216 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 29 times 0, 1, 39 times 0, 1, 48 times 0, 1, 55 times 0) [i] based on linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using
(117−29, 117, 196249)-Net in Base 4 — Upper bound on s
There is no (88, 117, 196250)-net in base 4, because
- 1 times m-reduction [i] would yield (88, 116, 196250)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6901 928765 047982 611475 272311 329100 802462 131107 579113 344151 549130 028376 > 4116 [i]