Best Known (18−11, 18, s)-Nets in Base 128
(18−11, 18, 300)-Net over F128 — Constructive and digital
Digital (7, 18, 300)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (1, 12, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128 (see above)
- digital (1, 6, 150)-net over F128, using
(18−11, 18, 309)-Net over F128 — Digital
Digital (7, 18, 309)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12818, 309, F128, 11) (dual of [309, 291, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(12818, 381, F128, 11) (dual of [381, 363, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(12818, 381, F128, 11) (dual of [381, 363, 12]-code), using
(18−11, 18, 386)-Net in Base 128 — Constructive
(7, 18, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- (2, 13, 257)-net in base 128, using
- 3 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 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
- base change [i] based on digital (0, 14, 257)-net over F256, using
- 3 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- digital (0, 5, 129)-net over F128, using
(18−11, 18, 299601)-Net in Base 128 — Upper bound on s
There is no (7, 18, 299602)-net in base 128, because
- 1 times m-reduction [i] would yield (7, 17, 299602)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 664623 480810 292981 312414 077268 716068 > 12817 [i]